Work all of the problems in this self-test without using a calculator. Then check your work by consulting the answers in the back of the book. Where weaknesses show up, use the reference that follows each answer to find the section in the test that provides the necessary review. 1. Replace each question mark with an appropriate expression that will illustrate the use of the indicated real number property: (A) Commutative (·): x ( y + z ) = ? (B) Associative (+): 2 + ( x + y ) = ? (C) Distributive: (2 + 3) x = ?

Question

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Answer 1

Textbook Question

Chapter DPT, Problem 1E

Work all of the problems in this self-test without using a calculator. Then check your work by consulting the answers in the back of the book. Where weaknesses show up, use the reference that follows each answer to find the section in the test that provides the necessary review.

1. Replace each question mark with an appropriate expression that will illustrate the use of the indicated real number property:

(A) Commutative (·): x(y + z) = ?

(B) Associative (+): 2 + (x + y) = ?

(C) Distributive: (2 + 3)x = ?

(A)

Expert Solution

To determine

To find: The equivalent expression of x(y+z) by using the commutative property.

Answer to Problem 1E

The expression x(y+z) is equivalent to (y+z)x.

Explanation of Solution

Given:

The expression is x(y+z).

Property used:

Commutative property:

“The order of factors does not affect the result of expression”.

That is, if x commutes with y under the operation "⋅", then x⋅y=y⋅x.

Illustration:

The factors of the given expression x(y+z) are x and (y+z).

Apply the commutative property by changing the order of factors to obtain the required equivalent expression.

That is, x(y+z) is altered to (y+z)x.

Therefore, the expression x(y+z) is equivalent to (y+z)x.

(B)

Expert Solution

To determine

To find: The equivalent expression of 2+(x+y) by using the associative property.

Answer to Problem 1E

The expression 2+(x+y) is equivalent to (2+x)+y.

Explanation of Solution

Given:

The expression is 2+(x+y).

Property used:

Associative property:

The grouping of any two numbers in an expression which contains more than two terms does affect the resulting expression.

That is, the property of associative under “+” operation states that (a+b)+c=a+(b+c).

Illustration:

Apply the associative property by changing the group of terms to obtain the required equivalent expression.

That is, 2+(x+y) is altered to (2+x)+y.

Therefore, the expression of 2+(x+y) is equivalent to (2+x)+y.

(C)

Expert Solution

To determine

To find: The equivalent expression of (2+3)x by using the distributive property.

Answer to Problem 1E

The expression (2+3)x is equivalent to 2x+3x.

Explanation of Solution

Given:

The expression is (2+3)x.

Property used:

Distributive property:

The property of left distributive under multiplication over addition state that (a+b)c=ac+bc.

Illustration:

Apply the left distributive property under multiplication over addition to obtain the required equivalent expression.

That is, (2+3)x is altered to 2x+3x.

Hence, the expression of (2+3)x equivalent to 2x+3x.

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03:09

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7. [10 marks] Let G = (V,E) be a 3-connected graph. We prove that for every x, y, z Є V, there is a cycle in G on which x, y, and z all lie. (a) First prove that there are two internally disjoint xy-paths Po and P₁. (b) If z is on either Po or P₁, then combining Po and P₁ produces a cycle on which x, y, and z all lie. So assume that z is not on Po and not on P₁. Now prove that there are three paths Qo, Q1, and Q2 such that: ⚫each Qi starts at z; • each Qi ends at a vertex w; that is on Po or on P₁, where wo, w₁, and w₂ are distinct; the paths Qo, Q1, Q2 are disjoint from each other (except at the start vertex 2) and are disjoint from the paths Po and P₁ (except at the end vertices wo, W1, and w₂). (c) Use paths Po, P₁, Qo, Q1, and Q2 to prove that there is a cycle on which x, y, and z all lie. (To do this, notice that two of the w; must be on the same Pj.)

Answer 2

Textbook Question

Chapter DPT, Problem 1E

Work all of the problems in this self-test without using a calculator. Then check your work by consulting the answers in the back of the book. Where weaknesses show up, use the reference that follows each answer to find the section in the test that provides the necessary review.

1. Replace each question mark with an appropriate expression that will illustrate the use of the indicated real number property:

(A) Commutative (·): x(y + z) = ?

(B) Associative (+): 2 + (x + y) = ?

(C) Distributive: (2 + 3)x = ?

(A)

Expert Solution

To determine

To find: The equivalent expression of x(y+z) by using the commutative property.

Answer to Problem 1E

The expression x(y+z) is equivalent to (y+z)x.

Explanation of Solution

Given:

The expression is x(y+z).

Property used:

Commutative property:

“The order of factors does not affect the result of expression”.

That is, if x commutes with y under the operation "⋅", then x⋅y=y⋅x.

Illustration:

The factors of the given expression x(y+z) are x and (y+z).

Apply the commutative property by changing the order of factors to obtain the required equivalent expression.

That is, x(y+z) is altered to (y+z)x.

Therefore, the expression x(y+z) is equivalent to (y+z)x.

(B)

Expert Solution

To determine

To find: The equivalent expression of 2+(x+y) by using the associative property.

Answer to Problem 1E

The expression 2+(x+y) is equivalent to (2+x)+y.

Explanation of Solution

Given:

The expression is 2+(x+y).

Property used:

Associative property:

The grouping of any two numbers in an expression which contains more than two terms does affect the resulting expression.

That is, the property of associative under “+” operation states that (a+b)+c=a+(b+c).

Illustration:

Apply the associative property by changing the group of terms to obtain the required equivalent expression.

That is, 2+(x+y) is altered to (2+x)+y.

Therefore, the expression of 2+(x+y) is equivalent to (2+x)+y.

(C)

Expert Solution

To determine

To find: The equivalent expression of (2+3)x by using the distributive property.

Answer to Problem 1E

The expression (2+3)x is equivalent to 2x+3x.

Explanation of Solution

Given:

The expression is (2+3)x.

Property used:

Distributive property:

The property of left distributive under multiplication over addition state that (a+b)c=ac+bc.

Illustration:

Apply the left distributive property under multiplication over addition to obtain the required equivalent expression.

That is, (2+3)x is altered to 2x+3x.

Hence, the expression of (2+3)x equivalent to 2x+3x.

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03:09

Answer 3

Textbook Question

Chapter DPT, Problem 1E

Work all of the problems in this self-test without using a calculator. Then check your work by consulting the answers in the back of the book. Where weaknesses show up, use the reference that follows each answer to find the section in the test that provides the necessary review.

1. Replace each question mark with an appropriate expression that will illustrate the use of the indicated real number property:

(A) Commutative (·): x(y + z) = ?

(B) Associative (+): 2 + (x + y) = ?

(C) Distributive: (2 + 3)x = ?

(A)

Expert Solution

To determine

To find: The equivalent expression of x(y+z) by using the commutative property.

Answer to Problem 1E

The expression x(y+z) is equivalent to (y+z)x.

Explanation of Solution

Given:

The expression is x(y+z).

Property used:

Commutative property:

“The order of factors does not affect the result of expression”.

That is, if x commutes with y under the operation "⋅", then x⋅y=y⋅x.

Illustration:

The factors of the given expression x(y+z) are x and (y+z).

Apply the commutative property by changing the order of factors to obtain the required equivalent expression.

That is, x(y+z) is altered to (y+z)x.

Therefore, the expression x(y+z) is equivalent to (y+z)x.

(B)

Expert Solution

To determine

To find: The equivalent expression of 2+(x+y) by using the associative property.

Answer to Problem 1E

The expression 2+(x+y) is equivalent to (2+x)+y.

Explanation of Solution

Given:

The expression is 2+(x+y).

Property used:

Associative property:

The grouping of any two numbers in an expression which contains more than two terms does affect the resulting expression.

That is, the property of associative under “+” operation states that (a+b)+c=a+(b+c).

Illustration:

Apply the associative property by changing the group of terms to obtain the required equivalent expression.

That is, 2+(x+y) is altered to (2+x)+y.

Therefore, the expression of 2+(x+y) is equivalent to (2+x)+y.

(C)

Expert Solution

To determine

To find: The equivalent expression of (2+3)x by using the distributive property.

Answer to Problem 1E

The expression (2+3)x is equivalent to 2x+3x.

Explanation of Solution

Given:

The expression is (2+3)x.

Property used:

Distributive property:

The property of left distributive under multiplication over addition state that (a+b)c=ac+bc.

Illustration:

Apply the left distributive property under multiplication over addition to obtain the required equivalent expression.

That is, (2+3)x is altered to 2x+3x.

Hence, the expression of (2+3)x equivalent to 2x+3x.

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03:09

Answer 4

Textbook Question

Chapter DPT, Problem 1E

Work all of the problems in this self-test without using a calculator. Then check your work by consulting the answers in the back of the book. Where weaknesses show up, use the reference that follows each answer to find the section in the test that provides the necessary review.

1. Replace each question mark with an appropriate expression that will illustrate the use of the indicated real number property:

(A) Commutative (·): x(y + z) = ?

(B) Associative (+): 2 + (x + y) = ?

(C) Distributive: (2 + 3)x = ?

(A)

Expert Solution

To determine

To find: The equivalent expression of x(y+z) by using the commutative property.

Answer to Problem 1E

The expression x(y+z) is equivalent to (y+z)x.

Explanation of Solution

Given:

The expression is x(y+z).

Property used:

Commutative property:

“The order of factors does not affect the result of expression”.

That is, if x commutes with y under the operation "⋅", then x⋅y=y⋅x.

Illustration:

The factors of the given expression x(y+z) are x and (y+z).

Apply the commutative property by changing the order of factors to obtain the required equivalent expression.

That is, x(y+z) is altered to (y+z)x.

Therefore, the expression x(y+z) is equivalent to (y+z)x.

(B)

Expert Solution

To determine

To find: The equivalent expression of 2+(x+y) by using the associative property.

Answer to Problem 1E

The expression 2+(x+y) is equivalent to (2+x)+y.

Explanation of Solution

Given:

The expression is 2+(x+y).

Property used:

Associative property:

The grouping of any two numbers in an expression which contains more than two terms does affect the resulting expression.

That is, the property of associative under “+” operation states that (a+b)+c=a+(b+c).

Illustration:

Apply the associative property by changing the group of terms to obtain the required equivalent expression.

That is, 2+(x+y) is altered to (2+x)+y.

Therefore, the expression of 2+(x+y) is equivalent to (2+x)+y.

(C)

Expert Solution

To determine

To find: The equivalent expression of (2+3)x by using the distributive property.

Answer to Problem 1E

The expression (2+3)x is equivalent to 2x+3x.

Explanation of Solution

Given:

The expression is (2+3)x.

Property used:

Distributive property:

The property of left distributive under multiplication over addition state that (a+b)c=ac+bc.

Illustration:

Apply the left distributive property under multiplication over addition to obtain the required equivalent expression.

That is, (2+3)x is altered to 2x+3x.

Hence, the expression of (2+3)x equivalent to 2x+3x.

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03:09

Answer 5

Textbook Question

Answer 6

Textbook Question

Answer 7

(A)

Expert Solution

To determine

To find: The equivalent expression of x(y+z) by using the commutative property.

Answer to Problem 1E

The expression x(y+z) is equivalent to (y+z)x.

Explanation of Solution

Given:

The expression is x(y+z).

Property used:

Commutative property:

“The order of factors does not affect the result of expression”.

That is, if x commutes with y under the operation "⋅", then x⋅y=y⋅x.

Illustration:

The factors of the given expression x(y+z) are x and (y+z).

Apply the commutative property by changing the order of factors to obtain the required equivalent expression.

That is, x(y+z) is altered to (y+z)x.

Therefore, the expression x(y+z) is equivalent to (y+z)x.

(B)

Expert Solution

To determine

To find: The equivalent expression of 2+(x+y) by using the associative property.

Answer to Problem 1E

The expression 2+(x+y) is equivalent to (2+x)+y.

Explanation of Solution

Given:

The expression is 2+(x+y).

Property used:

Associative property:

The grouping of any two numbers in an expression which contains more than two terms does affect the resulting expression.

That is, the property of associative under “+” operation states that (a+b)+c=a+(b+c).

Illustration:

Apply the associative property by changing the group of terms to obtain the required equivalent expression.

That is, 2+(x+y) is altered to (2+x)+y.

Therefore, the expression of 2+(x+y) is equivalent to (2+x)+y.

(C)

Expert Solution

To determine

To find: The equivalent expression of (2+3)x by using the distributive property.

Answer to Problem 1E

The expression (2+3)x is equivalent to 2x+3x.

Explanation of Solution

Given:

The expression is (2+3)x.

Property used:

Distributive property:

The property of left distributive under multiplication over addition state that (a+b)c=ac+bc.

Illustration:

Apply the left distributive property under multiplication over addition to obtain the required equivalent expression.

That is, (2+3)x is altered to 2x+3x.

Hence, the expression of (2+3)x equivalent to 2x+3x.

Answer 8

(A)

Expert Solution

To determine

To find: The equivalent expression of x(y+z) by using the commutative property.

Answer to Problem 1E

The expression x(y+z) is equivalent to (y+z)x.

Explanation of Solution

Given:

The expression is x(y+z).

Property used:

Commutative property:

“The order of factors does not affect the result of expression”.

That is, if x commutes with y under the operation "⋅", then x⋅y=y⋅x.

Illustration:

The factors of the given expression x(y+z) are x and (y+z).

Apply the commutative property by changing the order of factors to obtain the required equivalent expression.

That is, x(y+z) is altered to (y+z)x.

Therefore, the expression x(y+z) is equivalent to (y+z)x.

Answer 9

(A)

Expert Solution

Answer 10

(A)

Expert Solution

Answer 11

Expert Solution

Answer 12

To determine

To find: The equivalent expression of x(y+z) by using the commutative property.

Answer 13

Answer to Problem 1E

The expression x(y+z) is equivalent to (y+z)x.

Answer 14

Explanation of Solution

Given:

The expression is x(y+z).

Property used:

Commutative property:

“The order of factors does not affect the result of expression”.

That is, if x commutes with y under the operation "⋅", then x⋅y=y⋅x.

Illustration:

The factors of the given expression x(y+z) are x and (y+z).

Apply the commutative property by changing the order of factors to obtain the required equivalent expression.

That is, x(y+z) is altered to (y+z)x.

Therefore, the expression x(y+z) is equivalent to (y+z)x.

Answer 15

(B)

Expert Solution

To determine

To find: The equivalent expression of 2+(x+y) by using the associative property.

Answer to Problem 1E

The expression 2+(x+y) is equivalent to (2+x)+y.

Explanation of Solution

Given:

The expression is 2+(x+y).

Property used:

Associative property:

The grouping of any two numbers in an expression which contains more than two terms does affect the resulting expression.

That is, the property of associative under “+” operation states that (a+b)+c=a+(b+c).

Illustration:

Apply the associative property by changing the group of terms to obtain the required equivalent expression.

That is, 2+(x+y) is altered to (2+x)+y.

Therefore, the expression of 2+(x+y) is equivalent to (2+x)+y.

Answer 16

(B)

Expert Solution

Answer 17

(B)

Expert Solution

Answer 18

Expert Solution

Answer 19

To determine

To find: The equivalent expression of 2+(x+y) by using the associative property.

Answer 20

Answer to Problem 1E

The expression 2+(x+y) is equivalent to (2+x)+y.

Answer 21

Explanation of Solution

Given:

The expression is 2+(x+y).

Property used:

Associative property:

The grouping of any two numbers in an expression which contains more than two terms does affect the resulting expression.

That is, the property of associative under “+” operation states that (a+b)+c=a+(b+c).

Illustration:

Apply the associative property by changing the group of terms to obtain the required equivalent expression.

That is, 2+(x+y) is altered to (2+x)+y.

Therefore, the expression of 2+(x+y) is equivalent to (2+x)+y.

Answer 22

(C)

Expert Solution

To determine

To find: The equivalent expression of (2+3)x by using the distributive property.

Answer to Problem 1E

The expression (2+3)x is equivalent to 2x+3x.

Explanation of Solution

Given:

The expression is (2+3)x.

Property used:

Distributive property:

The property of left distributive under multiplication over addition state that (a+b)c=ac+bc.

Illustration:

Apply the left distributive property under multiplication over addition to obtain the required equivalent expression.

That is, (2+3)x is altered to 2x+3x.

Hence, the expression of (2+3)x equivalent to 2x+3x.

Answer 23

(C)

Expert Solution

Answer 24

(C)

Expert Solution

Answer 25

Expert Solution

Answer 26

To determine

To find: The equivalent expression of (2+3)x by using the distributive property.

Answer 27

Answer to Problem 1E

The expression (2+3)x is equivalent to 2x+3x.

Answer 28

Explanation of Solution

Given:

The expression is (2+3)x.

Property used:

Distributive property:

The property of left distributive under multiplication over addition state that (a+b)c=ac+bc.

Illustration:

Apply the left distributive property under multiplication over addition to obtain the required equivalent expression.

That is, (2+3)x is altered to 2x+3x.

Hence, the expression of (2+3)x equivalent to 2x+3x.

Answer 29

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Answer to Problem 1E

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