Let X and Y be sets. Let us say that a subset S of the cartesian product X x Y is a rectangle if S = A × B for some A C X and BCY. 10. (a) Prove that every one-element subset of X x Y is a rectangle. (b) Prove that whenever X is a set with more than one element, the subset {(x, æ) : ¤ € X} S = of X x X is not a rectangle. (Here Y = X.) (c) Is the union of two rectangles in X x Y always a rectangle? Give either a proof that this is true or an example to show that it is false.

Q: (c) Let X, Y and Z be sets. Prove that (X\Y)n (X \ Z) = X \ (YUZ).

A: We prove the given statement by using set theory.

Q: this question and subsequently, let a₁, A2,..., a5 denote these numbers: 2,2,1,7,3 Let X denote the…

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Q: 4. Let X be a set. Let A, B be the elements of the power set of X, P(A). Define the symmetric…

A: When we collect one or more well defined object and keep them together, it forms a set. The theory…

Q: Let A and B be sets. Prove that A U B = An B if and only if A = B

A: First show if A∪B=A∩B then A=B, then show that if A=B then we have A∪B=A∩B.

Q: Determine whether the set S = {1, x², 3 + x²} spans P2. O S spans P2. O s does not span P2.

A: Consider the given set S. S=1,x2,3+x2 Now, make the matrix from the given set. 103000011

Q: Let A, B, and C be sets (a) Prove that ACB iff A – B = Ø. (b) Prove that if A C BUCand ANB = Ø, then…

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Q: Let A, B, and C be sets. Prove (AUB)NC= (ANC) U(BNC)

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Q: prove for every two sets A and B that A=(A-B)U(A n B)

A: To prove: We want to prove  for every two sets A and B that A=A-B∪A∩B

Q: 2. Let X= [0, 10] be the universal set and A = [1,4], B = (2,8), C = [3, 6) be the subsets of X.…

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Q: Let A, B, C, D be sets. (a) (b) Prove that Ax (B \ C') = (A × B) \ (A × C). Prove that (A x B) n (C…

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Q: 9. Which of the following are true for all sets A, B, and C. If the statement is true, prove it by…

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Q: (3) Use an element proof to prove that for all sets A, B, C, and D, if An C =0 then (A x B)n (C x D)…

A: The proof of the given statement is given in the explanation part.Explanation:Steps and explanations…

Q: Let A and B be any sets. Show that: (a) A – B, B - A, and An B are disjoint with each other. (b) A =…

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Q: Let A and B be nonempty sets. Prove that A x B = B x A if and only if A = B.

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Q: (19) Prove or disprove ∀ sets A, B, and C: (A – B) − C = (A – C) – (B – C).

A: We will use the Venn diagram show the proof: Left side:

Q: II. Let Q-set of all real numbers, A-index set. Identify the generalized union and intersection of…

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Q: 2. Find a set A such that A = {2, JA]}.

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Q: (ANB) –C= (A –C)NB.

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Q: (iv) Prove that for any sets A, B,C we have A U (B nc) = (AU B) n (AUC)

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Q: (i) If A and B are two sets such that ACB, then m* (A) ≤m* (B).

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Q: A set F is called an Fo-set (or an Fo) if it is a countable union of closed sets. Likewise, a set G…

A: a) Take An=( -1/n , 1+ 1/n ) , which is open  Then taking intersection from n=1 to infinity we will…

Q: VII.1. Describe the set (R × Z) ∩ (Z × R)

A: We have to describe the set (R × Z) ∩ (Z × R).

Q: Q.No.2 Let X be a universal set and A, B be the subsets of X.Then prove the following. (a) ((A*…

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Q: 1. Let A = {1, 2} and let B = {x, y}. Write out the following sets by listing out all of their…

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Q: prove: for any two sets X and Y, #(X x Y) = #(X) • #(Y)

A: To prove it equals the product of the cardinality of both the sets.

Q: Prove that if X∩Y = X∩Z and X∪Y = X∪Z, then Y = Z for all sets X,Y, and Z.

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Q: 2. Prove Lemma: For all sets A, B, and C, A x (BnC) = (A × B)n (A × C)

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Q: The difference of two sets, A - B, is defined as the set containing all elements of A except those…

A: U={1,2,3,4,5,6,7,8,9} A={3,5,9} B={2,4,6,7,8,9} A-B=A∩B' Therefore, first find B'=?

Q: Write out the elements of the set 2X where X= = {1,{0}, Ø, {2,3}}.

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