Prove that the given sets are denumerable.

Question

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Answer 1

Question

Chapter 5.2, Problem 3E

a.

To determine

Prove that the given sets are denumerable.

a.

Expert Solution

Explanation of Solution

Given information:

D+ , the odd positive integers.

Calculation:

An infinite set is denumerable if it is equivalent to the set of natural number. We have given

D+ , the odd positive integers.

Now define the function f from N to D+,f(n)=2n−1 .

•f is one to onelet m and n be natural numbers such that f(m)=f(n)f(m)=f(n)⇒2m−1=2n−1⇒m=n

•f is onto D+let y∈D+ be given then for n=y+12f(n)=yf is bijective function. D+ is denumerable

Hence, proved.

b.

To determine

Prove that the given sets are denumerable.

b.

Expert Solution

Explanation of Solution

Given information:

3N , the positive integer multiples of 3 .

Calculation:

An infinite set is denumerable if it is equivalent to the set of natural number. We have given

3N , the positive integer multiples of 3 .

Now define the function f from N to D+,f(n)=2n−1 .

•f is one to one.let m and n be natural numbers such that f(m)=f(n)f(m)=f(n)⇒3m=3n⇒m=n

•f is onto 3Nlet y∈3N be given then for n=y3f(n)=yf is bijective function. D+ is denumerable.

Hence, proved.

c.

To determine

Prove that the given sets are denumerable.

c.

Expert Solution

Explanation of Solution

Given information:

3Z , the integer multiples of 3 .

Calculation:

An infinite set is denumerable if it is equivalent to the set of natural number. We have given

3Z , the integer multiples of 3 .

Now define the function f from N to 3Z,f(n)={3n/2 n is even3(1−n)/2 , n is odd .

•f is one to one.let m and n be natural numbers such that f(m)=f(n)f(m)=f(n)⇒3n/2=3(1−n)/2⇒2n=1

•f is onto 3Zlet y∈3Z be given then for n={2y3,y≥01−2y3,y<0f(n)=yf is bijective function. 3Z is denumerable.

Hence, proved.

d.

To determine

Prove that the given sets are denumerable.

d.

Expert Solution

Explanation of Solution

Given information:

{n:n∈N and n>6}

Calculation:

An infinite set is denumerable if it is equivalent to the set of natural number. We have given

{n:n∈N and n>6}

Now define the function f from N to {n:n∈N and n>6} f(n)=n+6 .

•f is one to one.let m and n be natural numbers such that f(m)=f(n)f(m)=f(n)⇒m+6=n+6⇒m=n

•f is onto {n:n∈N and n>6}let y∈{n:n∈N and n>6} be given then for n=y−6f(n)=yf is bijective function. {n:n∈N and n>6} is denumerable

Hence, proved.

e.

To determine

Prove that the given sets are denumerable.

e.

Expert Solution

Explanation of Solution

Given information:

{x:x∈Z and x<−12}

Calculation:

An infinite set is denumerable if it is equivalent to the set of natural number. We have given

{x:x∈Z and x<−12}

Now define the function f from N to {x:x∈Z and x<−12} , f(n)=−n−12 .

•f is one to one.let m and n be natural numbers such that f(m)=f(n)f(m)=f(n)⇒−m−12=−n−12⇒m=n

•f is onto {x:x∈Z and x<−12}let y∈{x:x∈Z and x<−12} be given then for n=−y−12f(n)=yf is bijective function. {x:x∈Z and x<−12} is denumerable

Hence, proved.

f.

To determine

Prove that the given sets are denumerable.

f.

Expert Solution

Explanation of Solution

Given information:

N−{5,6}

Calculation:

An infinite set is denumerable if it is equivalent to the set of natural number. We have given

N−{5,6}

Now define the function f from N to N−{5,6} , f(n)={n,n<5n+2, n≥5 .

•f is one to one.let m and n be natural numbers such that f(m)=f(n)f(m)=f(n)assume m=n+2this is contradiction⇒m=n

•f is onto N−{5,6}let y∈N−{5,6} be given then for n={y,y<5y−2,≥5f(n)=yf is bijective function. N−{5,6} is denumerable.

Hence, proved.

g.

To determine

Prove that the given sets are denumerable.

g.

Expert Solution

Explanation of Solution

Given information:

{(x,y)∈N×R:xy=1}

Calculation:

An infinite set is denumerable if it is equivalent to the set of natural number. We have given

{(x,y)∈N×R:xy=1}

Now define the function f from N to {(x,y)∈N×R:xy=1} , f(x)=(x,1x) .

•f is one to one.let m and n be natural numbers such that f(x)=f(y)f(x)=f(y)⇒(x,1x)=(y,1y)⇒x=y

•f is onto {(x,y)∈N×R:xy=1}let y∈{(x,y)∈N×R:xy=1} be given then for n=x,yf(n)=x,yf is bijective function. {(x,y)∈N×R:xy=1} is denumerable.

Hence, proved.

h.

To determine

Prove that the given sets are denumerable.

h.

Expert Solution

Explanation of Solution

Given information:

{x∈Z:x=1(mod5)}

Calculation:

An infinite set is denumerable if it is equivalent to the set of natural number. We have given

{x∈Z:x=1(mod5)}

Now define the function f from N to {x∈Z:x=1(mod5)} , f(k)=(5k+1) .

•f is one to one.let x and y be natural numbers such that f(x)=f(y)f(x)=f(y)⇒(x,1x)=(y,1y)⇒x=y

•f is onto {x∈Z:x=1(mod5)}let y∈{x∈Z:x=1(mod5)} be given then for n=x,yf(n)=x,yf is bijective function. {x∈Z:x=1(mod5)} is denumerable

Hence, proved.

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Answer 2

Question

Chapter 5.2, Problem 3E

a.

To determine

Prove that the given sets are denumerable.

a.

Expert Solution

Explanation of Solution

Given information:

D+ , the odd positive integers.

Calculation:

An infinite set is denumerable if it is equivalent to the set of natural number. We have given

D+ , the odd positive integers.

Now define the function f from N to D+,f(n)=2n−1 .

•f is one to onelet m and n be natural numbers such that f(m)=f(n)f(m)=f(n)⇒2m−1=2n−1⇒m=n

•f is onto D+let y∈D+ be given then for n=y+12f(n)=yf is bijective function. D+ is denumerable

Hence, proved.

b.

To determine

Prove that the given sets are denumerable.

b.

Expert Solution

Explanation of Solution

Given information:

3N , the positive integer multiples of 3 .

Calculation:

An infinite set is denumerable if it is equivalent to the set of natural number. We have given

3N , the positive integer multiples of 3 .

Now define the function f from N to D+,f(n)=2n−1 .

•f is one to one.let m and n be natural numbers such that f(m)=f(n)f(m)=f(n)⇒3m=3n⇒m=n

•f is onto 3Nlet y∈3N be given then for n=y3f(n)=yf is bijective function. D+ is denumerable.

Hence, proved.

c.

To determine

Prove that the given sets are denumerable.

c.

Expert Solution

Explanation of Solution

Given information:

3Z , the integer multiples of 3 .

Calculation:

An infinite set is denumerable if it is equivalent to the set of natural number. We have given

3Z , the integer multiples of 3 .

Now define the function f from N to 3Z,f(n)={3n/2 n is even3(1−n)/2 , n is odd .

•f is one to one.let m and n be natural numbers such that f(m)=f(n)f(m)=f(n)⇒3n/2=3(1−n)/2⇒2n=1

•f is onto 3Zlet y∈3Z be given then for n={2y3,y≥01−2y3,y<0f(n)=yf is bijective function. 3Z is denumerable.

Hence, proved.

d.

To determine

Prove that the given sets are denumerable.

d.

Expert Solution

Explanation of Solution

Given information:

{n:n∈N and n>6}

Calculation:

An infinite set is denumerable if it is equivalent to the set of natural number. We have given

{n:n∈N and n>6}

Now define the function f from N to {n:n∈N and n>6} f(n)=n+6 .

•f is one to one.let m and n be natural numbers such that f(m)=f(n)f(m)=f(n)⇒m+6=n+6⇒m=n

•f is onto {n:n∈N and n>6}let y∈{n:n∈N and n>6} be given then for n=y−6f(n)=yf is bijective function. {n:n∈N and n>6} is denumerable

Hence, proved.

e.

To determine

Prove that the given sets are denumerable.

e.

Expert Solution

Explanation of Solution

Given information:

{x:x∈Z and x<−12}

Calculation:

An infinite set is denumerable if it is equivalent to the set of natural number. We have given

{x:x∈Z and x<−12}

Now define the function f from N to {x:x∈Z and x<−12} , f(n)=−n−12 .

•f is one to one.let m and n be natural numbers such that f(m)=f(n)f(m)=f(n)⇒−m−12=−n−12⇒m=n

•f is onto {x:x∈Z and x<−12}let y∈{x:x∈Z and x<−12} be given then for n=−y−12f(n)=yf is bijective function. {x:x∈Z and x<−12} is denumerable

Hence, proved.

f.

To determine

Prove that the given sets are denumerable.

f.

Expert Solution

Explanation of Solution

Given information:

N−{5,6}

Calculation:

An infinite set is denumerable if it is equivalent to the set of natural number. We have given

N−{5,6}

Now define the function f from N to N−{5,6} , f(n)={n,n<5n+2, n≥5 .

•f is one to one.let m and n be natural numbers such that f(m)=f(n)f(m)=f(n)assume m=n+2this is contradiction⇒m=n

•f is onto N−{5,6}let y∈N−{5,6} be given then for n={y,y<5y−2,≥5f(n)=yf is bijective function. N−{5,6} is denumerable.

Hence, proved.

g.

To determine

Prove that the given sets are denumerable.

g.

Expert Solution

Explanation of Solution

Given information:

{(x,y)∈N×R:xy=1}

Calculation:

An infinite set is denumerable if it is equivalent to the set of natural number. We have given

{(x,y)∈N×R:xy=1}

Now define the function f from N to {(x,y)∈N×R:xy=1} , f(x)=(x,1x) .

•f is one to one.let m and n be natural numbers such that f(x)=f(y)f(x)=f(y)⇒(x,1x)=(y,1y)⇒x=y

•f is onto {(x,y)∈N×R:xy=1}let y∈{(x,y)∈N×R:xy=1} be given then for n=x,yf(n)=x,yf is bijective function. {(x,y)∈N×R:xy=1} is denumerable.

Hence, proved.

h.

To determine

Prove that the given sets are denumerable.

h.

Expert Solution

Explanation of Solution

Given information:

{x∈Z:x=1(mod5)}

Calculation:

An infinite set is denumerable if it is equivalent to the set of natural number. We have given

{x∈Z:x=1(mod5)}

Now define the function f from N to {x∈Z:x=1(mod5)} , f(k)=(5k+1) .

•f is one to one.let x and y be natural numbers such that f(x)=f(y)f(x)=f(y)⇒(x,1x)=(y,1y)⇒x=y

•f is onto {x∈Z:x=1(mod5)}let y∈{x∈Z:x=1(mod5)} be given then for n=x,yf(n)=x,yf is bijective function. {x∈Z:x=1(mod5)} is denumerable

Hence, proved.

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Answer 3

Question

Chapter 5.2, Problem 3E

a.

To determine

Prove that the given sets are denumerable.

a.

Expert Solution

Explanation of Solution

Given information:

D+ , the odd positive integers.

Calculation:

An infinite set is denumerable if it is equivalent to the set of natural number. We have given

D+ , the odd positive integers.

Now define the function f from N to D+,f(n)=2n−1 .

•f is one to onelet m and n be natural numbers such that f(m)=f(n)f(m)=f(n)⇒2m−1=2n−1⇒m=n

•f is onto D+let y∈D+ be given then for n=y+12f(n)=yf is bijective function. D+ is denumerable

Hence, proved.

b.

To determine

Prove that the given sets are denumerable.

b.

Expert Solution

Explanation of Solution

Given information:

3N , the positive integer multiples of 3 .

Calculation:

An infinite set is denumerable if it is equivalent to the set of natural number. We have given

3N , the positive integer multiples of 3 .

Now define the function f from N to D+,f(n)=2n−1 .

•f is one to one.let m and n be natural numbers such that f(m)=f(n)f(m)=f(n)⇒3m=3n⇒m=n

•f is onto 3Nlet y∈3N be given then for n=y3f(n)=yf is bijective function. D+ is denumerable.

Hence, proved.

c.

To determine

Prove that the given sets are denumerable.

c.

Expert Solution

Explanation of Solution

Given information:

3Z , the integer multiples of 3 .

Calculation:

An infinite set is denumerable if it is equivalent to the set of natural number. We have given

3Z , the integer multiples of 3 .

Now define the function f from N to 3Z,f(n)={3n/2 n is even3(1−n)/2 , n is odd .

•f is one to one.let m and n be natural numbers such that f(m)=f(n)f(m)=f(n)⇒3n/2=3(1−n)/2⇒2n=1

•f is onto 3Zlet y∈3Z be given then for n={2y3,y≥01−2y3,y<0f(n)=yf is bijective function. 3Z is denumerable.

Hence, proved.

d.

To determine

Prove that the given sets are denumerable.

d.

Expert Solution

Explanation of Solution

Given information:

{n:n∈N and n>6}

Calculation:

An infinite set is denumerable if it is equivalent to the set of natural number. We have given

{n:n∈N and n>6}

Now define the function f from N to {n:n∈N and n>6} f(n)=n+6 .

•f is one to one.let m and n be natural numbers such that f(m)=f(n)f(m)=f(n)⇒m+6=n+6⇒m=n

•f is onto {n:n∈N and n>6}let y∈{n:n∈N and n>6} be given then for n=y−6f(n)=yf is bijective function. {n:n∈N and n>6} is denumerable

Hence, proved.

e.

To determine

Prove that the given sets are denumerable.

e.

Expert Solution

Explanation of Solution

Given information:

{x:x∈Z and x<−12}

Calculation:

An infinite set is denumerable if it is equivalent to the set of natural number. We have given

{x:x∈Z and x<−12}

Now define the function f from N to {x:x∈Z and x<−12} , f(n)=−n−12 .

•f is one to one.let m and n be natural numbers such that f(m)=f(n)f(m)=f(n)⇒−m−12=−n−12⇒m=n

•f is onto {x:x∈Z and x<−12}let y∈{x:x∈Z and x<−12} be given then for n=−y−12f(n)=yf is bijective function. {x:x∈Z and x<−12} is denumerable

Hence, proved.

f.

To determine

Prove that the given sets are denumerable.

f.

Expert Solution

Explanation of Solution

Given information:

N−{5,6}

Calculation:

An infinite set is denumerable if it is equivalent to the set of natural number. We have given

N−{5,6}

Now define the function f from N to N−{5,6} , f(n)={n,n<5n+2, n≥5 .

•f is one to one.let m and n be natural numbers such that f(m)=f(n)f(m)=f(n)assume m=n+2this is contradiction⇒m=n

•f is onto N−{5,6}let y∈N−{5,6} be given then for n={y,y<5y−2,≥5f(n)=yf is bijective function. N−{5,6} is denumerable.

Hence, proved.

g.

To determine

Prove that the given sets are denumerable.

g.

Expert Solution

Explanation of Solution

Given information:

{(x,y)∈N×R:xy=1}

Calculation:

An infinite set is denumerable if it is equivalent to the set of natural number. We have given

{(x,y)∈N×R:xy=1}

Now define the function f from N to {(x,y)∈N×R:xy=1} , f(x)=(x,1x) .

•f is one to one.let m and n be natural numbers such that f(x)=f(y)f(x)=f(y)⇒(x,1x)=(y,1y)⇒x=y

•f is onto {(x,y)∈N×R:xy=1}let y∈{(x,y)∈N×R:xy=1} be given then for n=x,yf(n)=x,yf is bijective function. {(x,y)∈N×R:xy=1} is denumerable.

Hence, proved.

h.

To determine

Prove that the given sets are denumerable.

h.

Expert Solution

Explanation of Solution

Given information:

{x∈Z:x=1(mod5)}

Calculation:

An infinite set is denumerable if it is equivalent to the set of natural number. We have given

{x∈Z:x=1(mod5)}

Now define the function f from N to {x∈Z:x=1(mod5)} , f(k)=(5k+1) .

•f is one to one.let x and y be natural numbers such that f(x)=f(y)f(x)=f(y)⇒(x,1x)=(y,1y)⇒x=y

•f is onto {x∈Z:x=1(mod5)}let y∈{x∈Z:x=1(mod5)} be given then for n=x,yf(n)=x,yf is bijective function. {x∈Z:x=1(mod5)} is denumerable

Hence, proved.

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Answer 4

Question

Chapter 5.2, Problem 3E

a.

To determine

Prove that the given sets are denumerable.

a.

Expert Solution

Explanation of Solution

Given information:

D+ , the odd positive integers.

Calculation:

An infinite set is denumerable if it is equivalent to the set of natural number. We have given

D+ , the odd positive integers.

Now define the function f from N to D+,f(n)=2n−1 .

•f is one to onelet m and n be natural numbers such that f(m)=f(n)f(m)=f(n)⇒2m−1=2n−1⇒m=n

•f is onto D+let y∈D+ be given then for n=y+12f(n)=yf is bijective function. D+ is denumerable

Hence, proved.

b.

To determine

Prove that the given sets are denumerable.

b.

Expert Solution

Explanation of Solution

Given information:

3N , the positive integer multiples of 3 .

Calculation:

An infinite set is denumerable if it is equivalent to the set of natural number. We have given

3N , the positive integer multiples of 3 .

Now define the function f from N to D+,f(n)=2n−1 .

•f is one to one.let m and n be natural numbers such that f(m)=f(n)f(m)=f(n)⇒3m=3n⇒m=n

•f is onto 3Nlet y∈3N be given then for n=y3f(n)=yf is bijective function. D+ is denumerable.

Hence, proved.

c.

To determine

Prove that the given sets are denumerable.

c.

Expert Solution

Explanation of Solution

Given information:

3Z , the integer multiples of 3 .

Calculation:

An infinite set is denumerable if it is equivalent to the set of natural number. We have given

3Z , the integer multiples of 3 .

Now define the function f from N to 3Z,f(n)={3n/2 n is even3(1−n)/2 , n is odd .

•f is one to one.let m and n be natural numbers such that f(m)=f(n)f(m)=f(n)⇒3n/2=3(1−n)/2⇒2n=1

•f is onto 3Zlet y∈3Z be given then for n={2y3,y≥01−2y3,y<0f(n)=yf is bijective function. 3Z is denumerable.

Hence, proved.

d.

To determine

Prove that the given sets are denumerable.

d.

Expert Solution

Explanation of Solution

Given information:

{n:n∈N and n>6}

Calculation:

An infinite set is denumerable if it is equivalent to the set of natural number. We have given

{n:n∈N and n>6}

Now define the function f from N to {n:n∈N and n>6} f(n)=n+6 .

•f is one to one.let m and n be natural numbers such that f(m)=f(n)f(m)=f(n)⇒m+6=n+6⇒m=n

•f is onto {n:n∈N and n>6}let y∈{n:n∈N and n>6} be given then for n=y−6f(n)=yf is bijective function. {n:n∈N and n>6} is denumerable

Hence, proved.

e.

To determine

Prove that the given sets are denumerable.

e.

Expert Solution

Explanation of Solution

Given information:

{x:x∈Z and x<−12}

Calculation:

An infinite set is denumerable if it is equivalent to the set of natural number. We have given

{x:x∈Z and x<−12}

Now define the function f from N to {x:x∈Z and x<−12} , f(n)=−n−12 .

•f is one to one.let m and n be natural numbers such that f(m)=f(n)f(m)=f(n)⇒−m−12=−n−12⇒m=n

•f is onto {x:x∈Z and x<−12}let y∈{x:x∈Z and x<−12} be given then for n=−y−12f(n)=yf is bijective function. {x:x∈Z and x<−12} is denumerable

Hence, proved.

f.

To determine

Prove that the given sets are denumerable.

f.

Expert Solution

Explanation of Solution

Given information:

N−{5,6}

Calculation:

An infinite set is denumerable if it is equivalent to the set of natural number. We have given

N−{5,6}

Now define the function f from N to N−{5,6} , f(n)={n,n<5n+2, n≥5 .

•f is one to one.let m and n be natural numbers such that f(m)=f(n)f(m)=f(n)assume m=n+2this is contradiction⇒m=n

•f is onto N−{5,6}let y∈N−{5,6} be given then for n={y,y<5y−2,≥5f(n)=yf is bijective function. N−{5,6} is denumerable.

Hence, proved.

g.

To determine

Prove that the given sets are denumerable.

g.

Expert Solution

Explanation of Solution

Given information:

{(x,y)∈N×R:xy=1}

Calculation:

An infinite set is denumerable if it is equivalent to the set of natural number. We have given

{(x,y)∈N×R:xy=1}

Now define the function f from N to {(x,y)∈N×R:xy=1} , f(x)=(x,1x) .

•f is one to one.let m and n be natural numbers such that f(x)=f(y)f(x)=f(y)⇒(x,1x)=(y,1y)⇒x=y

•f is onto {(x,y)∈N×R:xy=1}let y∈{(x,y)∈N×R:xy=1} be given then for n=x,yf(n)=x,yf is bijective function. {(x,y)∈N×R:xy=1} is denumerable.

Hence, proved.

h.

To determine

Prove that the given sets are denumerable.

h.

Expert Solution

Explanation of Solution

Given information:

{x∈Z:x=1(mod5)}

Calculation:

An infinite set is denumerable if it is equivalent to the set of natural number. We have given

{x∈Z:x=1(mod5)}

Now define the function f from N to {x∈Z:x=1(mod5)} , f(k)=(5k+1) .

•f is one to one.let x and y be natural numbers such that f(x)=f(y)f(x)=f(y)⇒(x,1x)=(y,1y)⇒x=y

•f is onto {x∈Z:x=1(mod5)}let y∈{x∈Z:x=1(mod5)} be given then for n=x,yf(n)=x,yf is bijective function. {x∈Z:x=1(mod5)} is denumerable

Hence, proved.

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Answer 5

Chapter 5.2, Problem 3E

a.

To determine

Prove that the given sets are denumerable.

a.

Expert Solution

Explanation of Solution

Given information:

D+ , the odd positive integers.

Calculation:

An infinite set is denumerable if it is equivalent to the set of natural number. We have given

D+ , the odd positive integers.

Now define the function f from N to D+,f(n)=2n−1 .

•f is one to onelet m and n be natural numbers such that f(m)=f(n)f(m)=f(n)⇒2m−1=2n−1⇒m=n

•f is onto D+let y∈D+ be given then for n=y+12f(n)=yf is bijective function. D+ is denumerable

Hence, proved.

b.

To determine

Prove that the given sets are denumerable.

b.

Expert Solution

Explanation of Solution

Given information:

3N , the positive integer multiples of 3 .

Calculation:

An infinite set is denumerable if it is equivalent to the set of natural number. We have given

3N , the positive integer multiples of 3 .

Now define the function f from N to D+,f(n)=2n−1 .

•f is one to one.let m and n be natural numbers such that f(m)=f(n)f(m)=f(n)⇒3m=3n⇒m=n

•f is onto 3Nlet y∈3N be given then for n=y3f(n)=yf is bijective function. D+ is denumerable.

Hence, proved.

c.

To determine

Prove that the given sets are denumerable.

c.

Expert Solution

Explanation of Solution

Given information:

3Z , the integer multiples of 3 .

Calculation:

An infinite set is denumerable if it is equivalent to the set of natural number. We have given

3Z , the integer multiples of 3 .

Now define the function f from N to 3Z,f(n)={3n/2 n is even3(1−n)/2 , n is odd .

•f is one to one.let m and n be natural numbers such that f(m)=f(n)f(m)=f(n)⇒3n/2=3(1−n)/2⇒2n=1

•f is onto 3Zlet y∈3Z be given then for n={2y3,y≥01−2y3,y<0f(n)=yf is bijective function. 3Z is denumerable.

Hence, proved.

d.

To determine

Prove that the given sets are denumerable.

d.

Expert Solution

Explanation of Solution

Given information:

{n:n∈N and n>6}

Calculation:

An infinite set is denumerable if it is equivalent to the set of natural number. We have given

{n:n∈N and n>6}

Now define the function f from N to {n:n∈N and n>6} f(n)=n+6 .

•f is one to one.let m and n be natural numbers such that f(m)=f(n)f(m)=f(n)⇒m+6=n+6⇒m=n

•f is onto {n:n∈N and n>6}let y∈{n:n∈N and n>6} be given then for n=y−6f(n)=yf is bijective function. {n:n∈N and n>6} is denumerable

Hence, proved.

e.

To determine

Prove that the given sets are denumerable.

e.

Expert Solution

Explanation of Solution

Given information:

{x:x∈Z and x<−12}

Calculation:

An infinite set is denumerable if it is equivalent to the set of natural number. We have given

{x:x∈Z and x<−12}

Now define the function f from N to {x:x∈Z and x<−12} , f(n)=−n−12 .

•f is one to one.let m and n be natural numbers such that f(m)=f(n)f(m)=f(n)⇒−m−12=−n−12⇒m=n

•f is onto {x:x∈Z and x<−12}let y∈{x:x∈Z and x<−12} be given then for n=−y−12f(n)=yf is bijective function. {x:x∈Z and x<−12} is denumerable

Hence, proved.

f.

To determine

Prove that the given sets are denumerable.

f.

Expert Solution

Explanation of Solution

Given information:

N−{5,6}

Calculation:

An infinite set is denumerable if it is equivalent to the set of natural number. We have given

N−{5,6}

Now define the function f from N to N−{5,6} , f(n)={n,n<5n+2, n≥5 .

•f is one to one.let m and n be natural numbers such that f(m)=f(n)f(m)=f(n)assume m=n+2this is contradiction⇒m=n

•f is onto N−{5,6}let y∈N−{5,6} be given then for n={y,y<5y−2,≥5f(n)=yf is bijective function. N−{5,6} is denumerable.

Hence, proved.

g.

To determine

Prove that the given sets are denumerable.

g.

Expert Solution

Explanation of Solution

Given information:

{(x,y)∈N×R:xy=1}

Calculation:

An infinite set is denumerable if it is equivalent to the set of natural number. We have given

{(x,y)∈N×R:xy=1}

Now define the function f from N to {(x,y)∈N×R:xy=1} , f(x)=(x,1x) .

•f is one to one.let m and n be natural numbers such that f(x)=f(y)f(x)=f(y)⇒(x,1x)=(y,1y)⇒x=y

•f is onto {(x,y)∈N×R:xy=1}let y∈{(x,y)∈N×R:xy=1} be given then for n=x,yf(n)=x,yf is bijective function. {(x,y)∈N×R:xy=1} is denumerable.

Hence, proved.

h.

To determine

Prove that the given sets are denumerable.

h.

Expert Solution

Explanation of Solution

Given information:

{x∈Z:x=1(mod5)}

Calculation:

An infinite set is denumerable if it is equivalent to the set of natural number. We have given

{x∈Z:x=1(mod5)}

Now define the function f from N to {x∈Z:x=1(mod5)} , f(k)=(5k+1) .

•f is one to one.let x and y be natural numbers such that f(x)=f(y)f(x)=f(y)⇒(x,1x)=(y,1y)⇒x=y

•f is onto {x∈Z:x=1(mod5)}let y∈{x∈Z:x=1(mod5)} be given then for n=x,yf(n)=x,yf is bijective function. {x∈Z:x=1(mod5)} is denumerable

Hence, proved.

Answer 6

Chapter 5.2, Problem 3E

a.

To determine

Prove that the given sets are denumerable.

a.

Expert Solution

Explanation of Solution

Given information:

D+ , the odd positive integers.

Calculation:

An infinite set is denumerable if it is equivalent to the set of natural number. We have given

D+ , the odd positive integers.

Now define the function f from N to D+,f(n)=2n−1 .

•f is one to onelet m and n be natural numbers such that f(m)=f(n)f(m)=f(n)⇒2m−1=2n−1⇒m=n

•f is onto D+let y∈D+ be given then for n=y+12f(n)=yf is bijective function. D+ is denumerable

Hence, proved.

Answer 7

Chapter 5.2, Problem 3E

a.

To determine

Prove that the given sets are denumerable.

Answer 8

a.

Expert Solution

Explanation of Solution

Given information:

D+ , the odd positive integers.

Calculation:

An infinite set is denumerable if it is equivalent to the set of natural number. We have given

D+ , the odd positive integers.

Now define the function f from N to D+,f(n)=2n−1 .

•f is one to onelet m and n be natural numbers such that f(m)=f(n)f(m)=f(n)⇒2m−1=2n−1⇒m=n

•f is onto D+let y∈D+ be given then for n=y+12f(n)=yf is bijective function. D+ is denumerable

Hence, proved.

Answer 9

a.

Expert Solution

Answer 10

a.

Expert Solution

Answer 11

Expert Solution

Answer 12

b.

To determine

Prove that the given sets are denumerable.

b.

Expert Solution

Explanation of Solution

Given information:

3N , the positive integer multiples of 3 .

Calculation:

An infinite set is denumerable if it is equivalent to the set of natural number. We have given

3N , the positive integer multiples of 3 .

Now define the function f from N to D+,f(n)=2n−1 .

•f is one to one.let m and n be natural numbers such that f(m)=f(n)f(m)=f(n)⇒3m=3n⇒m=n

•f is onto 3Nlet y∈3N be given then for n=y3f(n)=yf is bijective function. D+ is denumerable.

Hence, proved.

Answer 13

b.

To determine

Prove that the given sets are denumerable.

Answer 14

b.

Expert Solution

Explanation of Solution

Given information:

3N , the positive integer multiples of 3 .

Calculation:

An infinite set is denumerable if it is equivalent to the set of natural number. We have given

3N , the positive integer multiples of 3 .

Now define the function f from N to D+,f(n)=2n−1 .

•f is one to one.let m and n be natural numbers such that f(m)=f(n)f(m)=f(n)⇒3m=3n⇒m=n

•f is onto 3Nlet y∈3N be given then for n=y3f(n)=yf is bijective function. D+ is denumerable.

Hence, proved.

Answer 15

b.

Expert Solution

Answer 16

b.

Expert Solution

Answer 17

Expert Solution

Answer 18

c.

To determine

Prove that the given sets are denumerable.

c.

Expert Solution

Explanation of Solution

Given information:

3Z , the integer multiples of 3 .

Calculation:

An infinite set is denumerable if it is equivalent to the set of natural number. We have given

3Z , the integer multiples of 3 .

Now define the function f from N to 3Z,f(n)={3n/2 n is even3(1−n)/2 , n is odd .

•f is one to one.let m and n be natural numbers such that f(m)=f(n)f(m)=f(n)⇒3n/2=3(1−n)/2⇒2n=1

•f is onto 3Zlet y∈3Z be given then for n={2y3,y≥01−2y3,y<0f(n)=yf is bijective function. 3Z is denumerable.

Hence, proved.

Answer 19

c.

To determine

Prove that the given sets are denumerable.

Answer 20

c.

Expert Solution

Explanation of Solution

Given information:

3Z , the integer multiples of 3 .

Calculation:

An infinite set is denumerable if it is equivalent to the set of natural number. We have given

3Z , the integer multiples of 3 .

Now define the function f from N to 3Z,f(n)={3n/2 n is even3(1−n)/2 , n is odd .

•f is one to one.let m and n be natural numbers such that f(m)=f(n)f(m)=f(n)⇒3n/2=3(1−n)/2⇒2n=1

•f is onto 3Zlet y∈3Z be given then for n={2y3,y≥01−2y3,y<0f(n)=yf is bijective function. 3Z is denumerable.

Hence, proved.

Answer 21

c.

Expert Solution

Answer 22

c.

Expert Solution

Answer 23

Expert Solution

Answer 24

d.

To determine

Prove that the given sets are denumerable.

d.

Expert Solution

Explanation of Solution

Given information:

{n:n∈N and n>6}

Calculation:

An infinite set is denumerable if it is equivalent to the set of natural number. We have given

{n:n∈N and n>6}

Now define the function f from N to {n:n∈N and n>6} f(n)=n+6 .

•f is one to one.let m and n be natural numbers such that f(m)=f(n)f(m)=f(n)⇒m+6=n+6⇒m=n

•f is onto {n:n∈N and n>6}let y∈{n:n∈N and n>6} be given then for n=y−6f(n)=yf is bijective function. {n:n∈N and n>6} is denumerable

Hence, proved.

Answer 25

d.

To determine

Prove that the given sets are denumerable.

Answer 26

d.

Expert Solution

Explanation of Solution

Given information:

{n:n∈N and n>6}

Calculation:

An infinite set is denumerable if it is equivalent to the set of natural number. We have given

{n:n∈N and n>6}

Now define the function f from N to {n:n∈N and n>6} f(n)=n+6 .

•f is one to one.let m and n be natural numbers such that f(m)=f(n)f(m)=f(n)⇒m+6=n+6⇒m=n

•f is onto {n:n∈N and n>6}let y∈{n:n∈N and n>6} be given then for n=y−6f(n)=yf is bijective function. {n:n∈N and n>6} is denumerable

Hence, proved.

Answer 27

d.

Expert Solution

Answer 28

d.

Expert Solution

Answer 29

Expert Solution

Answer 30

e.

To determine

Prove that the given sets are denumerable.

e.

Expert Solution

Explanation of Solution

Given information:

{x:x∈Z and x<−12}

Calculation:

An infinite set is denumerable if it is equivalent to the set of natural number. We have given

{x:x∈Z and x<−12}

Now define the function f from N to {x:x∈Z and x<−12} , f(n)=−n−12 .

•f is one to one.let m and n be natural numbers such that f(m)=f(n)f(m)=f(n)⇒−m−12=−n−12⇒m=n

•f is onto {x:x∈Z and x<−12}let y∈{x:x∈Z and x<−12} be given then for n=−y−12f(n)=yf is bijective function. {x:x∈Z and x<−12} is denumerable

Hence, proved.

Answer 31

e.

To determine

Prove that the given sets are denumerable.

Answer 32

e.

Expert Solution

Explanation of Solution

Given information:

{x:x∈Z and x<−12}

Calculation:

An infinite set is denumerable if it is equivalent to the set of natural number. We have given

{x:x∈Z and x<−12}

Now define the function f from N to {x:x∈Z and x<−12} , f(n)=−n−12 .

•f is one to one.let m and n be natural numbers such that f(m)=f(n)f(m)=f(n)⇒−m−12=−n−12⇒m=n

•f is onto {x:x∈Z and x<−12}let y∈{x:x∈Z and x<−12} be given then for n=−y−12f(n)=yf is bijective function. {x:x∈Z and x<−12} is denumerable

Hence, proved.

Answer 33

e.

Expert Solution

Answer 34

e.

Expert Solution

Answer 35

Expert Solution

Answer 36

f.

To determine

Prove that the given sets are denumerable.

f.

Expert Solution

Explanation of Solution

Given information:

N−{5,6}

Calculation:

An infinite set is denumerable if it is equivalent to the set of natural number. We have given

N−{5,6}

Now define the function f from N to N−{5,6} , f(n)={n,n<5n+2, n≥5 .

•f is one to one.let m and n be natural numbers such that f(m)=f(n)f(m)=f(n)assume m=n+2this is contradiction⇒m=n

•f is onto N−{5,6}let y∈N−{5,6} be given then for n={y,y<5y−2,≥5f(n)=yf is bijective function. N−{5,6} is denumerable.

Hence, proved.

Answer 37

f.

To determine

Prove that the given sets are denumerable.

Answer 38

f.

Expert Solution

Explanation of Solution

Given information:

N−{5,6}

Calculation:

An infinite set is denumerable if it is equivalent to the set of natural number. We have given

N−{5,6}

Now define the function f from N to N−{5,6} , f(n)={n,n<5n+2, n≥5 .

•f is one to one.let m and n be natural numbers such that f(m)=f(n)f(m)=f(n)assume m=n+2this is contradiction⇒m=n

•f is onto N−{5,6}let y∈N−{5,6} be given then for n={y,y<5y−2,≥5f(n)=yf is bijective function. N−{5,6} is denumerable.

Hence, proved.

Answer 39

f.

Expert Solution

Answer 40

f.

Expert Solution

Answer 41

Expert Solution

Answer 42

g.

To determine

Prove that the given sets are denumerable.

g.

Expert Solution

Explanation of Solution

Given information:

{(x,y)∈N×R:xy=1}

Calculation:

An infinite set is denumerable if it is equivalent to the set of natural number. We have given

{(x,y)∈N×R:xy=1}

Now define the function f from N to {(x,y)∈N×R:xy=1} , f(x)=(x,1x) .

•f is one to one.let m and n be natural numbers such that f(x)=f(y)f(x)=f(y)⇒(x,1x)=(y,1y)⇒x=y

•f is onto {(x,y)∈N×R:xy=1}let y∈{(x,y)∈N×R:xy=1} be given then for n=x,yf(n)=x,yf is bijective function. {(x,y)∈N×R:xy=1} is denumerable.

Hence, proved.

Answer 43

g.

To determine

Prove that the given sets are denumerable.

Answer 44

g.

Expert Solution

Explanation of Solution

Given information:

{(x,y)∈N×R:xy=1}

Calculation:

An infinite set is denumerable if it is equivalent to the set of natural number. We have given

{(x,y)∈N×R:xy=1}

Now define the function f from N to {(x,y)∈N×R:xy=1} , f(x)=(x,1x) .

•f is one to one.let m and n be natural numbers such that f(x)=f(y)f(x)=f(y)⇒(x,1x)=(y,1y)⇒x=y

•f is onto {(x,y)∈N×R:xy=1}let y∈{(x,y)∈N×R:xy=1} be given then for n=x,yf(n)=x,yf is bijective function. {(x,y)∈N×R:xy=1} is denumerable.

Hence, proved.

Answer 45

g.

Expert Solution

Answer 46

g.

Expert Solution

Answer 47

Expert Solution

Answer 48

h.

To determine

Prove that the given sets are denumerable.

h.

Expert Solution

Explanation of Solution

Given information:

{x∈Z:x=1(mod5)}

Calculation:

An infinite set is denumerable if it is equivalent to the set of natural number. We have given

{x∈Z:x=1(mod5)}

Now define the function f from N to {x∈Z:x=1(mod5)} , f(k)=(5k+1) .

•f is one to one.let x and y be natural numbers such that f(x)=f(y)f(x)=f(y)⇒(x,1x)=(y,1y)⇒x=y

•f is onto {x∈Z:x=1(mod5)}let y∈{x∈Z:x=1(mod5)} be given then for n=x,yf(n)=x,yf is bijective function. {x∈Z:x=1(mod5)} is denumerable

Hence, proved.

Answer 49

h.

To determine

Prove that the given sets are denumerable.

Answer 50

h.

Expert Solution

Explanation of Solution

Given information:

{x∈Z:x=1(mod5)}

Calculation:

An infinite set is denumerable if it is equivalent to the set of natural number. We have given

{x∈Z:x=1(mod5)}

Now define the function f from N to {x∈Z:x=1(mod5)} , f(k)=(5k+1) .

•f is one to one.let x and y be natural numbers such that f(x)=f(y)f(x)=f(y)⇒(x,1x)=(y,1y)⇒x=y

•f is onto {x∈Z:x=1(mod5)}let y∈{x∈Z:x=1(mod5)} be given then for n=x,yf(n)=x,yf is bijective function. {x∈Z:x=1(mod5)} is denumerable

Hence, proved.

Answer 51

h.

Expert Solution

Answer 52

h.

Expert Solution

Answer 53

Expert Solution

Answer 54

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Prove that the given sets are denumerable.

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Prove that the given sets are denumerable.

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Prove that the given sets are denumerable.

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Prove that the given sets are denumerable.

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Prove that the given sets are denumerable.

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Prove that the given sets are denumerable.

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Prove that the given sets are denumerable.

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Prove that the given sets are denumerable.

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Prove that the given sets are denumerable.

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Chapter 5 Solutions