3. Let A, B, C, and D be sets. (a) Prove that (A × C) n (B × D) = (An B) × (Cn D).

Q: (avb)л (aVc) d → e (bA c) → (F → k) ^ c) .. -F

A:

Q: Prove that (ab + cd)² ≤ (a² + c²)(b² + ď²) for all a, b, c, d ɛ R.

A:

Q: Let A, B, and C are three sets, prove that Cx (A – B) = (C x A) – (C × B) C.

A:

Q: C. Show that both R and R² may be covered by countably many open balls.

A: Firstly we have shown that R may be covered by countably many open balls. Let us  consider the set…

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: If B C A, then (i) P(AB)=P(A) - P(B), (ii) P(B) ≤ P(A)

A:

Q: (а b)с + (а - с)b + (с b)a 3 ve that v belongs to B.

A:

Q: If A and B are independent, and the P(A) = 0.35 and P(B) = 0.45 find: a) P(A ∩ B)…

A: Given that P(A) = 0.35 P(B) = 0.45 A, B are independent events.

Q: Let S1= {1, 5, 10, 15, 20} and let S2 = {1, 2, 3, 4, 5}. What is: (a) S1 ∪ S2 (b) S1 ∩ S2 (c) S1 –…

A: Given: The given sets are S1=1, 5, 10, 15, 20 and S2=1, 2, 3, 4, 5. To find: The main aim is to find…

Q: Let f AB, prove the following: (a) if S, T are subsets of A, then f(SNT) ≤ f(S) nf(T); (b) if U, V…

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: 3. Is this sentence consistent? “There is someone who loves exactly those people who do not love…

A:

Q: · (b + c) = ² 7. Prove that à. = a · b + ac for all a, b, c € R³.

A:

Q: Ema thinks that (u) xw, exists. Paul disagrees with her. Who is right? Explain why.

A: Dot product gives a scalar value  .

Q: 1) What is the sum of 8y 1- 2y? and -3y2 + 2y + 7?

A:

Q: 5. Let A 3D, В %3 and C %3 Find a) (A · B)C b) А : (2С) c) ||B||2

A:

Q: Prove that |A U B| + |A n B| = |A| + |B|

A: Let A and B are two sets. A = number of elements in the set A. B = number of elements in the set B.…

Q: Suppose that P(A) = 0.1, P(B) = 0.3, and P(A|B) + P(B|A) = 0.8. Find P(A ∩ B).

A: From the information, given thatThe formula of the conditional probability, for any two events…

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

A: Prove the Cauchy-Schwartz inequality…

Q: O Is the set S=\[1.2], [-4 2. abaris for R?.

A: We know dimension of ℝ2  is 2  this implies basis of ℝ2 contains two linearly independent vectors.…

Q: If P (B) = 1/4, P (A ∪ B) = 1/2 and P (A|B) = 2/3, then which of the following statements is true?…

A: GivenP(B)=14P(A∪B)=12P(A|B)=23

Q: 2. Let A = and B = al. Show that det(A + B) = det A + det B if and only if a + d = 0.

A:

Q: (a) If A₁ = {-i, -(i − 1), . . .−3,−2,−1, 0, 1, 2 }, for i = 1, 2, 3,. . . find (i) Un A6 and (ii)…

A: "Since you have asked multiple questions, we will solve the first question for you. If you want any…

Q: Show that both R and R² may be covered by countably many open balls.

A:

Q: Let P(C|D) = 0.2, P(D) = 0.3 and P(C|D) = 0.4, then C and D are Select one: a. disjoint b. not…

A:

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: Y D Prox

A: Product of two sets

Q: (3) (i) Determine the number of positive solutions of the equation x1+x2+x3+X4 = 14, where each x;…

A: Thank you for your question!! But according to the guidelines I will be answering one question only.…

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

A:

Q: Which of the answers below is the ordering of these running times from fastest to slowest? a)…

A: We have to arrange the following sequences in order from fastest running times to slowest. (a)…

Q: (m) W ) Given A, Be {sets}, V AC B, A – B= A.

A: Given: A,B∈sets and A⊂B since A⊂B⇒A-B=ϕ we have to prove A-B=A So, these will happen when A=ϕ So,…

Q: 6. show that the set A = {P= (₁4₁2) (R³ | 1²4 + √²2+x2 is an open out.

A:

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) ?

Q: true. Show that (a v bc)(a Abc); but the converse is not

A: Solution:

Q: Suppose that P ( A ) = 0.4,  P ( B ) = 0.5 and A , B are disjoint. Compute P ( A c ∪ B c ) .

A: From the provided information, P (A) = 0.4, P (B) = 0.5 The events A and B are disjoint.

Q: 7. Establish for A is m × n and B, C are n x p, that A(B+C) = AB + AC

A:

Q: 9. Given u =(2,-2), v = (5,8), and w = (-4, 3), find each of the following. (a) u v (b) (u - v)w (c)…

A:

Q: Evaluate f(e* +2*) dx a.) 2e* ++C b.) e2* + 2ln* + C c.) e* ++C d) e2x + 2ln x + C Evaluate fesx-2…

A: This question is based on indefinite integral.

Q: e independent, and the P(A) = 0.4 and P(B) = 0.6 find: a) P(A ∩ B)                  b)P(A U

A: P(A) = 0.4 P(B) = 0.6 to find a) P(AnB) b) P(AuB)