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

Q: Let Р and q be points in R³, then p∙q ≤ ||p||||q||. Hint: (a1, a2, a3) · (b₁,b2, b3) = a1b₁ + a2b2 +…

A: This is a problem of functional analysis.

Q: 11. For sets A and B, define AAB (A\B) U (B\A). Prove that A B if and only if AAB=0. - (a) Let A and…

A:

Q: For sets A and B, prove that AU (B – A) = B

A:

Q: Prove that the set {a E R: a2 not equal to 6} is open.

A:

Q: A W C с 3 р a 6.0 D g h f B

A:

Q: 5. Let A = { a, b, c }. Draw digraphs to represent relations on A that are: a) antisymmetric and…

A:

Q: Which of the following is always true about the Cartesian product of sets A and B? A. (A x B) x C =…

A:

Q: Prove: For all sets A, B and C, (AUB) - (CA) = AU (B-C).

A:

Q: Let A, B, and C be sets. 1. Prove: If A C B then ANC C BnC. 2. Disprove: If AnC CBNC, then A C B. 3.…

A:

Q: 4. For the four pairs of sets below, determine whether or not they are equipollent. i.) {2, 5, 10}…

A: We have to check the given sets are equipollent or not.

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: Let A= {k: k ∈ N, k ≤ 20}, B= {3k−1: k ∈ N}, and C= {2k+1: k ∈ N} 1. A INTERSECTION B INTERSECTION…

A:

Q: d) Suppose X = {1,2, 3} and Y = {5,6}. The number of elements of the set P(X× Y) =

A: Elements in X = 3. Elements in Y = 2 Elements in X × Y = 3 × 2 = 6

Q: Let A, B, and C be sets. Is the following equality true or false? A x (BUC) = (A × B) U (A × C')

A: Sets, distributive law

Q: Q17D. Define the sets A, B, C, and D as shown. Find (A ∩ C) ∪ (B ∩ D).

A:

Q: 4. Let A and B be sets. (a) If A and B are subsets of the same universal set U (so A × B is a subset…

A: The definition of complement is given at the end of proof

Q: Given the sets C = {1, -3} and D = {2, 3, 6}, find all ordered pairs (x,y)and those that satisfy the…

A:

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).

A:

Q: Consider the relations given below: xRy:x loves y xRy:x2y iii. xRy:x is in the same classroom with y…

A: If x = y and y = x then we can say that relation is symmetric relation.

Q: Let à (2, 4, 3) and b = (1, 3, – 4). = Find a b. . -

A: Let a→=2, 4, -3 and b→=1, 3, -4.To find a→·b→.

Q: Let (pos S) be the set of all positive combinations of the points of S. Which of the following…

A: We need to find out which statements are correct. To do that, we shall independently analyze both…

Q: Use any method you choose to show that An (B UC) = A U B n C, when A, B, and C are sets.

A:

Q: 18. Show that P→ qand-q→→p are logically equivalent.

A: We have to show that  and  are logically equivalent.

Q: Let A, B, C be sets. Prove that if C ⊆A, then A −(B −C) = (A −B) ∪C. Analogously, for any real…

A:

Q: Challenge. Establish the Cauchy-Schwartz inequality: For real numbers we have (a + az + ..+ a) (b+…

A: Prove the Cauchy-Schwartz inequality…

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

A:

Q: 3. Let A, B, C be sets. Provide a proof or a counterexample for each of the following statements. a.…

A:

Q: Let A and B be sets. Prove that A Ū B = (A − B) U (B − A) U (ANB).

A: Let A and B be any two sets.Prove that: .

Q: If  A ∩ B = ∅, P(A) = 0.4, P(A ∪ B) = 0.8. Find P(B).

A: Given : A∩B = ∅PA = 0.4PA∪B = 0.8

Q: (c) A \ B = A \ (An B). (d) A = (ANB) U (A \ B).

A:

Q: Find the complement of the set given that  U = {x | x I and −3 ≤ x ≤ 7}. (Enter your answers as a…

A: First we write U in roster form.

Q: -1 3 4 2 2 and s = 29 17 Let u = 0 6 3 1 3 4 18 60 s can be written as a linear combination of u,v…

A: