Let A and B be arbitrary sets. For each statement below, decide whether it is true or false. If the statement is true, prove it rigorously. If the statement is false, give a counterexample. (a) X CAN B if and only if X ≤ A and X C B. (b) X CAUB if and only if XCA or X CB.

Q: For all sets A and B, determine whether each of the statement is true or false.

A: For the given problem(a)  to check it is true or false.For this part we use the counter example to…

Q: Thoorem: is not closed set

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Q: For the predicate P(x, y) ::= [x + 2 = y], choose all the pairs of x and y such that P(x, y) is…

A: Given query is to find the correct option for the expression.

Q: If R is an integer domain, show that it contains exactly two elements a that satisfy a? = a.

A: Given R is an integral domain. We have to show that it contains exactly two elements a that satisfy…

Q: 1. The following subsets A, B, and C of R XR are relations on R. Sketch the set. (i) B = {(x, y):…

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Q: Find the complement of the set given that U = {x | x I and −3 ≤ x ≤ 9}. (Enter your answers as a…

A: First we write U and the set in roaster form.

Q: Prove or disprove: if x is in set B and x > inf(B), then x/2  is in set B.

A: Let B= The set of natural number then  InfB=1 Let x=3 Since x∈B & x>infB.

Q: Let A and B be sets. For each statement below, determine if it is true or false. If true, prove the…

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Q: Write the following in set-builder notation. All positive real numbers.

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Q: 1. Let A = [1, 3], B = (2, 5], and C= {2}. Write the inequalities (or equalities) that specify each…

A: We will just use of definitions of different type of sets

Q: For Exercise, write the set as a single interval. [(-0, 6) U (10, o)] N [8, 12)

A: We will draw them on a number line. Draw:

Q: Let D₁ = {x E RI-i ≤x≤i} = [-i,i] for all nonnegative integers i. (If i = 0, then Do = {0}.)…

A: The difference of two sets A and B is the set those elements which are in A but not in B .

Q: Let A and B be non-empty sets. Using the element method prove that: If A C B, then AU (B – A) = B.…

A: Given that, A and B are non-empty sets and set A is the subset of set B. To prove :

Q: Find the complement of the set given that  U = {x | x  I and −3 ≤ x ≤ 9}.  (Enter your answers as a…

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Q: We know that X*H=H if and only if XEH .Give an example to satisfy this statement

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Q: Use any method you choose to shc are sets.

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Q: Find the complement of the set given that  U = {x | x  I and −3 ≤ x ≤ 5}.  (Enter your answers as a…

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Q: Find the complement of the set given that  U = {x | x  I and −3 ≤ x ≤ 10}.  (Enter your answers as a…

A: We Know that,  If U is a universal set and A be any subset of U then the complement of A is the set…

Q: Let A and B be two sets. We denote A- B = { x | x∊A and xÏB}. Prove that A∪B=(A−B)∪(A∩B)∪(B−A).

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Q: 2. For all sets X, Y and Z, determine whether each of the statement is true or false. If it is true,…

A: (a) False. We know that if a set A has cardinality n , then it's power set P(A) has cardinality 2n…

Q: Suppose x ∈ R. Let p(x): x² - 2x - 15 < 17 then the statement x is (valid or invalid)?

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Q: Prove or disprove: If A, B, and C are sets, then A × (B − C) = (A × B) – (A × C).

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Q: Assume that A, B, C and D are sets. Verify that (A x B) U (C x D) ≤ (AUC) x (BUD).

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

A: we have to prove the lemma .

Q: Give an example of numbers a and b such that |a + b| &lt; |a|+|b|.

A: To give an example of numbers a and b such that

Q: Let A and B be sets. The symmetric difference of A and B, denoted A∆B, is the set A∆B = (A −B) ∪(B…

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Q: (−4, 5]∩[1, 9]. Write your answer in either set-builder or interval notation.

A: Here, we have to determine (−4, 5]∩[1, 9].