1. Provide an example of two sets A and B such that A ‡ B and both AU B and An B are countable. j. Provide an example of two sets A and B such that AUB is denumerable but A - B is finite and nonempty. k. Find examples to satisfy each of the following: i. f. Z→ Ris 1to1 but not onto. ii. g. Z→ N is onto but not 1to1. iii. b: N→N is neither 1to1 nor onto.

Q: contrapositive and converse of each statement. (a) If all birds fly, and if Tweety is a bird, then…

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Q: 1. (a) Consider the set S {n(-1)" : n e N}. Verify the existence or otherwise of Min(S) and Max(S).…

A: given that (1) (a)                                    S=n(-1)n:n∈N=2k(-1)2k:n=2k,…

Q: Let A and B be sets. Prove that A ≤ B in each of the following. (i) A = {x²: x € Z} and B = Z (ii) A…

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Q: 2. Decide whether each of the following sets are finite, denumerable or uncountable. Explain each of…

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Q: 1. f¹ B-A. 2. f is bijective. Suppose f: A→ B. Prove that the following statements are equivalent.

A: A bijection, also known as a bijective function, one-to-one correspondence, or invertible function,…

Q: 3. Let H≤ G. Then the number of left cosets of H in G is [G:H]. *True *False

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Q: Let x E A. Prove that A is an infinite set if and only if A≈ (A - {x}).

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Q: 1. (a) Consider the set S = {n(-1)" : n e N}. Verify the existence or otherwise of Min(S) and…

A: a) The given set is,S=n-1n:n∈ℕ

Q: J Assume E₂ and E₂ one measurable. Without using the fort that the Union of measurable sets in…

A: Given that  and  are measurable.We need to prove that  is measurable.We know that .We know that, if…

Q: (b) Consider the set Prove that inf A = 0. A = { n m² : n,m € Z¹, m≤n}.

A: Our objective is given below:

Q: Suppose S and T are nonempty sets of real numbers such that for x in S and y in T, we have x < y.…

A: Upper bound: Let A be a non-empty set. Then, a real number s is said to be the upper boundof the set…

Q: Let S=X belongs to R: x> =1. a) Show that the set S has lower bound, but no upper bounds. b) Show…

A: The answer is given as follows:

Q: Consider two sets, A = {1, 2, 3, ..., 100} and B = {3, 5, 7, 9, ... , 201}. Prove that |A| = |B| by…

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Q: 1. Let X be a nonempty set of sets. Prove that P(UX) = U{P(A)| A e X} if and only if UX E X.

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Q: Ο b. (ΧΔΥ)ΔΖ - ΧΔ(ΥΔΖ)

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Q: 5. Let A = {x E Z|x = 12r - 5 for some integer r} and B = {y € Z|y=6s +1 for some integer s}. Prove…

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Q: Which of the followings is a base of the set W = {[x 0 y] : x,y E R}:

A: As per given by the question, There are given that, Then, Find the base of the set W. Now, From the…

Q: Suppose A 1, .., An are sets and each f : A A i+1 is surjective. Prove or disprove: fn1° fn2 *** •…

A: To prove  the given function surjective we will use the fact that each f(i) is surjective

Q: Find an expression for sup in terms of inf E, sup E, inf F, sup F and prove it. 2F

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Q: 3. Let G = Q3. Find the right cosets of H in G for: a) H = ( K). b) H = (-1).

A: Let G=Q8. We need to find the right cosets of &lt;K&gt; and &lt;-1&gt;.

Q: Let :Q-21 → be an epimorphism. Hand written plz a) Show that f(n)-n for every n e Z. b) Deduce that,…

A: Here f:ℚ→ℚ is an epimorphism. That is f is a surjective ring homomorphism from ℚ to  ℚ. Now we can…

Q: Ex 4.) Let Sbe a non empty bounded set in 1. aj Let azo, and let a S:= {as: SES}. Prove that inf…

A: Let S be any -empty bounded set in a) Let  , and let  Since,   is a bounded subset of , Sup and Inf…

Q: Exercise. Consider the set R?. For every r = (x1, x2) E R? and y = (Yı, 42) E R², write P1(x, y) =|…

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Q: (#4) Show that a set SCR" is closed if and only if bdry(S) CS.

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Q: If Do, D1,..., Dn... nEN, are dense open sets of reals, then the intersection D = 0 Dn is dense in…

A: Condition: Dn is the open dense sets of reals, for all n∈ℕ. To find: Intersection of infinite open…

Q: 6. Let z E C. Let C be set of points z satisfying Im(z) 0 and Im Prove that C = {ZEC : |Z| = 1}.…

A: Given that  Im1+z+z21-z+z2=0 We know that if Imz=0 then Im1z=0 Therefore  Im1-z+z21+z+z2=0 Finding…

Q: Which of the following sets is closed in (R, Tu)? O a. { ne N}U{2} 72 2n O b. Q-N c. O c. [2,5) U…

A: Determine the meaning of ℝ, τu. Now, go with each option and check which is correct.

Q: Let A and B be any two sets. Prove that Ax B Bx A. Hint: Define f : AxB - Bx A by (a, b) (b, a) for…

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Q: Let S be an open subset in C and f EC¹(2). Let R be a closed rectangle with vertices v₁ = a +ic,…

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Q: The statement form (ac) ⇒ (bc) is equivalent to O [(~avc) ^ (av~c)] ^ [(~b v c) ^ (b v~c)] ~[(~a v…

A: Since you have posted a multiple question according to guildlines I will solve first question for…

Q: a) Supposef:A→N is a function such that for every k∈N, there are exactly two elements x in A such…

A: Let  f:A→ℕ be a function. For every k∈ℕ, there are exactly two elements x in A such that f(x)=k. For…

Q: Let S := a + √2 b + √2 : a, b EN Prove that S is countably infinite.

A: To prove that the set S defined as S={a+2b+2|a,b∈N} is countably infinite, need to establish a…

Q: 3. Prove that the image of a measurable set E under the translation 4 is measurable and m(E+a)…

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Q: Exercise 6. Let E be an infinite non-demumerable set. For every A € P(E), we set m(A) = 0 if A is…

A: Let E be an infinite non-denumerable set. m will be a measure on PE if it is non-negative, which…

Q: Let A, B be bounded subsets of R. Let A+B = {r+y:1 € A, y E B} and AB = {ry : 1 € A, y € B}. Which…

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Q: は F-R m Show that u is 3. sauomphic to F

A: V is isomorphic to F3 means  There is a bijection map between V to F3 such that that bijection is…

Q: 4. Let X be an ordered set, z E X, and SC X. Prove that x = sup S, if and only if Vy EX (yy).

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Q: 1. Let H ≤ G. Then by a right coset of H in G we mean a subset of G of the form: aH = {ah: heH, aeG}…

A: 1. Let H≤G. Then by a right coset of H in G we mean a subset of G of the form: aH=ah : h∈H, a∈G. To…