Exercises. Prove the following: 1. A ∧ (B ∧ C) ⊢ C 2. A ∧ (B → C), A → B ⊢ C 3. A ∧ (B ∧ C) ⊢ (A ∧ B) ∧ C 4. A, A → B ⊢ A ∧ B 5. A ∧ B, B → C ⊢ A ∧ (B ∧ C) 6. A → B ∧ C, A ∧ D ⊢ B ∧ C 7. A → C ⊢ A ∧ B → C ∧ B 8. A ∧ B → C ⊢ A → (B → C

Q: (2c) Prove or disprove the following statement: “ A UBCANB for any A and B "

A:

Q: 2. Which of the following operations is associative on Z? A. a*b = a b C. a*b = a B. a*b = max (a,…

A: Note: As per Bartleby guidelines if a question have more than 3 subparts and specific part is not…

Q: -8 -2 -9 6 4 4 0 4 w is in Col A. Is w in Nul A? Let A = 8 and w= 27 -2 Determine if

A:

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

A:

Q: Give an example where A, B ⊆ X and |A| = |B| but |X ― A| ≠ |X ― B|.

A: This cannot be possible if we take X to be finite set so we have to think an example where X is…

Q: 3+2i Evaluate 4-i

A: The given complex number is 3+2i4-i. The conjugate of the denominator is 4+i.

Q: If a+b c +d 4 [c- d с —d а — Ь 10 2 | find a, b, c, and d. If

A: Concept used: If two matrix are equal then their order and corresponding element are identical.…

Q: Let A = {1, 2}, B = {1, 2, 3, 4}, C = {5, 6} and D = {5, 6, 7, 8}. Verify that(i) A × (B ∩ C) = (A ×…

A: i) Answer(i) A × (B ∩ C) = (A × B) ∩ (A × C) (proved)

Q: Theorem 5.25. Let A, B, and C be sets. Then 1. A× (B N C) = (A × B) N (A × C) 2. A × (B U C) = (A ×…

A:

Q: Let a, b € N. Choose all correct statements. O gcd(-a, b) = gcd(a, -b) O ged(2a – b, a + b) =…

A:

Q: 5. Аx (В — С) %3D (А х В) — (А хС) 6. (А - В) хС %3 (АxC) — (В хС)

A: To prove the given statements.

Q: Check that P ˅(Q ˅ R) is equal to ┐(R ˄ Q ˄ P) → P

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: True or False?

A:

Q: 1) If u(x, t) satisfies the wave equation du Ozu, , prove the identity (1) u(x + h, t + k) + u(x –…

A:

Q: 6. For A, B, subscts of R, define A+B= {a+b:a€ A, bE B} and A B= (ab:a € A,bE B} (a) For A=…

A:

Q: Let U = {x | x is the name of one of the months in a year} J = {x | x  U and x begins with the…

A: Given that U = {x | x is the name of one of the months in a year} J = {x | x U and x begins with the…

Q: Let u = (2, -2, 3), v = (1, -3, 4), and w=(3, 6, -4). Evaluate the following 1. ||3u-5v+w||…

A:

Q: d) The set E = (-1,0) U (0, 1] is an open set. %3D

A: Given set is E = (-1, 0) ∪ (0, 1]

Q: Find P(C1)  l. Find P(V4)  m. Find P(C1 ∩∩ V4)

A: As per guidelines we are allowed to solve maximum of 3 sub parts If you need help with other…

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: find AB, BA, A +B, A - B, A, B, 5A, 3B. Observe that AB + BA. Show that (A - B)(A + B) # (A + B)(A –…

A:

Q: Let Ai = = [i, i + 1], i = 1, 2, 3, Then A₁ A2 A3 A4 is. a. {1,2,3,4,5} b. (1,5) d. [1,5] ....

A: We have given that Ai=[i,i+1]A1=[1,2]A2=[2,3]A3=[3,4]A4=[4,5]

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: If a|b, then a|(bc). Direct proof.

A:

Q: Show that ∗ : R × R → R given by           a ∗ b → a + 2b is not associative.

A: Given:Show that ∗: R × R → R given by           a ∗ b → a + 2b is not associative.

Q: T T T T F 18. Let A and B be sets then A + B if, and only if A ¢ B and B ¢ A. For sets A, B, and C,…

A:

Q: I. TRUE OR FALSE. Write 1 if the statement is always true. Otherwise write 0. 8. For any sets A, B,…

A:

Q: Let a, b,c € R4. Prove that c+a a+b b+c c+b a +c b+a Al

A: a,b,c∈ℝ+ab+bc+ca≥c+ac+b+a+ba+c+b+cb+a