I. Prove the following statements using the indicated proof. 1.) Let 1 € Z. If z is odd, then (1² +3)²021 is even. (Direct Proof)

Q: a – b -| < 1, if|a| < 1, and |b| < 1. 1- āb - Show that|-

A: From the given information. For , then show that .

Q: 4.) Find the value(s) of k such that A is singular. 1 k 2 A = -2 -k 3 1 -4

A: The given matrix is                               A=  1  k    2-2  0 -k   3  1 -4 We have to find…

Q: 6. Let An = [−3⁄, 1 + ¹). Find ♂ Añ and justify (not prove) that your answer is correct. n=1

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Q: 1. For each of the following statements: if the statement is true, then give a proof; if the…

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Q: only Q3: It is given that ε = {x: x is an integer, 1 ≤ x16} A = (x:x is a perfect square} B = {x:x…

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Q: If p times tne pth term of an A.P. is equal to q times the qth term of the same AP., then how that…

A: Given that If p times the pth term of an A. P. is equal to q times the qth term of same A. P.  To…

Q: Formalize the following statement: lim. AL lim sinx f 0≤ q<1 There is a program that gives the…

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Q: 2. ) For each statement, Determine if it is True. If it is true then prove it. If it is False then…

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Q: 10. Show that È k(C) = n2"-1. Using Combinatorial Proof. k=1

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Q: For example, if I wanted to prove this statement using a direct proof, I would start as follows: Va,…

A: This question is related to Number Theory

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A: 13 c. Given: m and n are two integers, where n&gt;2.

Q: Is the set {3m + v3ni|m, n E Z, b|m – n} the normal subgroup of the (C, +)group?

A: given :

Q: Discrete Mathematics: 3. Prove: 7n -1 is divisible by 3 for n ³ 0

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Q: show that: (For real nu

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Q: Let A= {X: X is an integer and r24}then A= {X: X is an integer and X<4} {X: X e N and X<4} a {X: X…

A: A detailed solution is given below. Z --» the set of integers.

Q: I. Prove the following statement using the indicated proof. Let r € Z. If r is odd, then (r2 +3)(r²…

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Q: Consider the following statement, for all x E Z: If x + 1 is even, then 3x² - 4 is odd. The correct…

A: A statement is given as "If x+1 is even, then 3x2-4 is odd". Also a way of proving the statement…

Q: This are a Math discrete structure questions.

A: Negation of If A, then B is A and not B A or B is not A and not B A ^ B is ⌐A v ⌐B

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A: We know that, Statement: If P then Q. Contrapositive: If ~Q then ~P.

Q: For all n, d EZ where d + 0, d\n n = dk for some kEZ. Disprove: For all integers a, 6, and c, if…

A: I have disproved the result

Q: 2. Prove (using direct proof): If 3a - 1 is even, then (a+2) is odd. Warning: Do not use fractions.…

A: We have to prove the given argument.

Q: 8. Suppose a, b e Z. Prove that a = b (mod 10) if and only if a = b (mod 2) and a = b (mod 5).

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Q: 2. Prove the following statement using a proof by contradiction. Explicitly indicate the assumptions…

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Q: п(п - 1)(п +1) 8. Prove that Ek(k +1) = 3 k=1

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Q: 6. Let A = {nk Z, n = 6k+5} and B = {nn is odd}. Prove that AC B.

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Q: Use the formula n+1` 1m + 2m +... + n™ = Σ k! k %3D k+1, k=1 to derive the identity: 14 + 24 + . +…

A: We have to derive the identity 14+24+.....+n4=nn+12n+13n2+3n−130 Use the formula…

Q: 14. Give a geometric explanation of why 2π/n 1-2 0 r² do = лr² 2 for any positive real number r and…

A: n2∫02πnr2dθ=πr2 For any positive real number r and any positive integer n. Prove this geometrically.…

Q: Question 29. Suppose a, b e N. If gcd(a, b) > 1, then b| a or b is not prime.

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Q: S. Given: AD BC LA= LB Prove: BD = AC D Proof: Statement Reasen 1. 1. 2. 2. 3. 3. 4. S.

A: Use the congruent postulate of the triangles.

Q: Detailed

A: We prove first part of Que 4 by Contradiction method.  Given that a is any integer. Any integer of…

Q: Prove the following statements: i. Let x, y and z be integers. If exactly one of x,y and z is odd…

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Q: i- n(n+1) i= 2 I (2i – 1) = n²

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Q: 4. 习 1) Complete the proof by filling in the statements or reasons appropriately. Rea Statements 1.…

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