(4) State which of the following statements are true, vacuously true, or false. (a) (A-B)' = A' - B'. (b) If n is an integer which is both even and odd, then n is equal to 0. (c) If P(A) {0} then A Ø.

Q: Compute each of the following. b. (3)* d. (-3). f. (1)** a. (- 10) - (" 10)° C. 50 147 e. g. 4+3 5-2…

A: Since you have asked multiple question, we will solve the first 3 question for you as per our guide…

Q: M(A) = 1875, n (B) = 2003, 1875-333 =1542 ➡n (ANB) Now ü) 2 m (A-B) 2 (B-A) = 333 5 a) The Venn…

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Q: 2 - If A and X then - 11 5 7 А.Х —

A: From the given problem:   A=-20-115 ; X=-17   Now,   A.X=-20-115.-17

Q: - 25 -dr (x – 4)(x + 1) Evaluate

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Q: 6. Evaluate In (x - In (r-...))dr in terms of some new variable t (do not simplify).

A: Given :- ∫lnx-lnx-...dx To find the given integral in terms of some new variables t

Q: The small positive integer n for which (1+i)=(1-1)" is =. ........ 8 4 12

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Q: . Show that p(X) = XP-1 +... +X +1 when p is prime.

A: For every integer n≥1, the nth cyclotomic polynomial Φn is the monic polynomial whose complex roots…

Q: If a = 2a - a² and b =1-(1-a), show that a = b for all natural numbers n. Assume that a, is a fixed…

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Q: A8. (a) State (without proof) the Intermediate Value Theorem. (b) Use the Intermediate Value Theorem…

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Q: 5. (*) If a and b are nonzero integers such that a | b and b|a, prove that a = ±b.

A: .

Q: 5. Prove the following identity for n 2 1: Fzn+1 = F+1+ F;

A: We need to prove that ,  F2n + 1 = F2n+1 + F2n We know that ,   Fm+n = FmFn+1 + Fm-1Fn ,  m ≥ 1 , n…

Q: Need

A: Given that, a can be written as base 10 x= a0+10a1+ ......+10nar We know that, If a divides b…

Q: (c) (i) Express the decimal 9.999 in the form where a, b € Z.

A:

Q: Solve the following ODE y +2y +y = x + e O a. (C1+ C2) e* + (x-2) + e /9 O b. (C1 + x C2) e* + (x-2)…

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Q: Prove that A2=I.

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Q: Let (a) te list eompeebee tolate E , , , () Repeat part a) fo 1 . -a. - e) Ina print statest, tate…

A: We will find  g(2+h) -g(2)h for all values of h and then conclude .  Note : maximum 3 subpart are…

Q: 2 Find (5a" + 4a*) da 1

A: We can find the integration value as below

Q: Let a and b be non-negative real numbers with b+ 0 and p,q > 1 and+= 1, then a show that ab <+

A: Define a function y : [0,∞)→ℝ by yt=1q+tp-t1p, for every t∈ [0,∞). Now differentiating,…

Q: 1. (a) 2 Let = (1, 0, -2) and = (-4, 2, 3). Find a real number x such that (ট' –æè) ⊥ à. (b)-1 (c) 3…

A: Let .To find a real number  such that .

Q: (3c) Let n E Z. Prove that if n is odd, then 4|(n² – 1)

A: here is the proof

Q: 9. Prove the following algebraic identity: (a - b)(a? + ab +b?)=a' -b'

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Q: (i) Deduce that the difference in the number of integers in the numerator and denominator of (* 2N N…

A: Properties of floor functions will be used.

Q: Find the dual of the Boolean expression I +1.0+ Fyz O z. (z+1). (7+ y+z) O T. (7 +0). (7 + y+z) O z.…

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Q: (12) (a) Prove that if x € Z and r² is even, then x is even. (b) State and prove an analogous…

A: We prove following statements by using contrapositive method.

Q: Ex 3.6.2 Suppose p is a prime, (a, p) = p° and (b, p) = p. Find (a + b, p*).

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Q: -1 - 1 Suppose A Then (A") = 2 1 ||

A: Here, the given matrix is: A=-1-121 To find (AT)-1. Note that (AT)-1=(A-1)T

Q: The dual of the following Boolean expression is (0 +1)(1+ yz) + 7 O (1.7) + (0.y + 2). z O (1.7) +…

A: We have to find the dual of 0+x1+yz+x

Q: Let Find formulas for the entries of M", where n is a positive integer. Hint: a formula such as 5…

A: The given problem is to find the entries of Mn for the given following matrix M. We have to find the…

Q: If xER then |1– x| = |x+1|- True O False

A: We have check given statement is true or not

Q: (:)? /3 9. Which of the following is equal to (b) (–1)"-12·5., 3"n! 3.5-7..... (2n + 1) (Зп — 4)…

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Q: (9) Let P, Q, R be (a) (¬P VQ) (b) (P mathematical statements. Negate the following statements.…

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Q: Write the following expressions in a more compact way (i.e., using index i, j as above). 1. C11 C2x2…

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Q: 5. Prove that the following are irrational: E-2-3', £-12¬!.

A: Expand the given functions and then determine their values. If the values are irrational then the…

Q: 1.f a" da + C, for any real number n. n+1 O A. True ов. False

A: The solution is given below

Q: (-1)" (n+ 2) 2n+1(n + 1) Compute the exact value of

A: