Let A, B, and C be sets. Show that a) (A ∪ B) ⊆ (A ∪ B ∪ C). b) (A ∩ B ∩ C) ⊆ (A ∩ B). c) (A − B) − C ⊆ A − C. d) (A − C) ∩ (C − B) = ∅. e) (B − A) ∪ (C − A) = (B ∪ C) − A.

Q: Give direct and indirect proofs of: p→q,¬r→¬q,¬r⇒¬p. p→¬q, ¬r→q, p⇒r. a∨b, c∧d, a→¬c⇒b.

A: Direct proof :  step  reason 1. p→q premise  2.¬p→¬q premise  3. ¬r premise  4. ¬q 2,3,MP…

Q: Match the sets to equal sets:  (Ac∪B)c(Ac∪B)c  A∩(B∪C)A∩(B∪C)  (A∪B)c(A∪B)c  A∪(B∩C)A∪(B∩C)…

A: Solution

Q: Find JA|, |B|, AB, and |AB|. 1 4 2 -3 0 A = -2 -4 1 B = 1 -1 3 3 1 2 11 (a) JA| (b) |B| (с) АВ (d)…

A: Find the determinant

Q: (2 3. Let A = G) and B = (; ). Compute and compare: %3D a. Det(A + B), det(A) + det(B). b. det(3A),3…

A: We have A=2-135 and B=1211 Part(a) Now if we have a matrix M=abcd, then, detM=abcd=ad-bc Therefore,…

Q: Let a, b, q, r E Z and assume that a and b are not both zero and b and r are not both zero. Show…

A:

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: a) Prove that (A∪B)c=Ac∩Bc. b) Prove thatA∪(B∩C) = (A∪B)∩(A∪C). c) Prove that ifA⊆B, thenA∩C⊆B∩C.…

A:

Q: Solve this proof A∪(B∩C)=(A∪B)∩(A∪C)

A: Step 1:We have to show that, for three sets A,B and C:A∪(B∩C)=(A∪B)∩(A∪C)the above rule is called…

Q: Given that A and B are independent with P(A ∪ B) = 0.8 and P(B c ) = 0.3, find P(A).

A:

Q: a) If a2 is even, then a is even. (b) If a|b and a|c, then a|(b − c). (c) If a|b and c|d, then…

A:

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: let A, B ⊂ R, and suppose A and B are closed. Prove that A ∩B is closed. Hint: You might try showing…

A:

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: . Consider the statement: Let A, B ∈ Z. If 6|A and 10|B, then 4|(A2 + B2). (a) Prove the statement.…

A: 5. Consider the statement: Let A, B ∈ Z. If 6|A and 10|B, then 4|(A^2 + B^2). (a) Prove the…

Q: If |A|  8 and |B|  5, what is |A ∪ B| + |A ∩ B| ?

A:

Q: (f) Let A, B and C be three sets. Prove by contradiction that if An BC C and r E B, then x ¢ A - C.

A: Let A, B and C be the three sets. Prove that, A∩B⊆C, and x∈B, then x∉A-C

Q: Let a,b be two different points in R^n one. We notice that d=(a,b)>0 (why is that so ?). Prove that…

A:

Q: Fill out the Venn diagram using the given information. |D|=23|D|=23 |E|=36|E|=36 |F|=31|F|=31…

A: Given information D  = 23 E = 36  F = 31 D∩E = 13 D∩F = 12 E∩F  = 21 D∩E∩F = 6 U = 52

Q: Let A={2,y,z} and B={3,x,z}. How many distinct elements are in (A×B)∩(B×A)

A: Given, A={2,y,z} B={3,x,z} How many distinct elements are in (A×B)∩(B×A) ?

Set Theory

Every field in the present world involves a set. The theory of sets was developed by German mathematician Georg Cantor while working on the “problems of trigonometric series”.