Prove the following statements. b. Let a, b EZ, then ab is odd if and only if both a and b are odd. c. Let a, b € Z. If a² + b² is odd then ab is even. d. Let a, b e Z. The sum a+bis even if and only if a and b have the same parity. e. Let a, b € Z, then a + b and a - b have the same parity.

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: . Let a, b € Z so that 4a + 5b is a multiple of 13. Prove that 13 divides 9a + 8b.

A: We have to show that 13 divides 9a+8b To prove this we using 4a+5b divisible by 13

Q: = [a b] and B =[2³]. a. Compute AB. b. If AB = 1-5₂ 1. Let A = find the entries of A, i.e. find the…

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Q: Prove or Disprove: Va, b, c E N, if a | bc and c| ab then b lac. (Here | denotes divides.)

A: We will disprove the given statement by giving a counter example that is we will give values for a…

Q: Let an = an-1+ an-2 with ao = 0, a1 = 1. Show that an is even iff ( if and only if) n is a multiple…

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Q: Let a, b and c be integers. Prove that if a does not divide b and a|(b+c), then a does not divide c

A: Let a, b and c be integers.

Q: Given A and B = [6 –7 -1 8], find AB and BA . | -2 АВ BA : ||

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Q: Let a, b, and c be natural numbers. Prove a ∙ (b + c) = a ∙b + a ∙ c.

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Q: a. Let a Z, then a² - 5a +7 is odd.

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Q: | − x + 2| − 3 ≤4

A: Given  -x+2-3 ≤ 4 We have to find the values of x.

Q: Which statement is an example of a transitive relationship? If x = 2y and 2y = 8, then x = 4. If clm…

A: Transitive relationship is a relationship in which if from the first two condition 3rd condition can…

Q: let a and b be integers. Show: if a * b is odd, then a is odd and b is odd

A: A number is even, if it is divisible by 2. A number is odd, if it is not divisible by 2. The odd…

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

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

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

A: Prove the Cauchy-Schwartz inequality…

Q: Recall that each permutation can be written as a product of disjoint cycles. How many permutations…

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Q: 1. The greatest common divisor c, of a and b, denoted as c = gcd(a, b), is the largest number that…

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Q: Suppose that P(A) = 0.4, P(B) = 0.3 and PAUB)-042, then a. A and B are dependent b. none of the…

A: We have to find out correct answer...

Q: Find the product of x and y if x, 3/2, 6/7, y  are in G.P

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Q: 9. Classify each of the following statements as true or false, for all whole numbers a, b, and c,…

A: ?Two integer a|b ( a divides b) means there exists an integer k such that       b= a×k . ?Product,…

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

A: