1. Prove that for all sets A and B and n € Z, n ≥ 1, the following holds: n (a) U(A₁\B): i=1 = n C i=1 A₁ \ B; n (b) (4₁\B) Ai i=1 n = (1₁) \₁ i=1 B.

Q: contrapositive and converse of each statement. (a) If all birds fly, and if Tweety is a bird, then…

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Q: 1. (a) Consider the set S {n(-1)" : n e N}. Verify the existence or otherwise of Min(S) and Max(S).…

A: given that (1) (a)                                    S=n(-1)n:n∈N=2k(-1)2k:n=2k,…

Q: b. Let E = [0, 1] ~ P. Then µ*E = 1, and if A N [0, 1] is a measur- able set, then m*(A N E) = mE.

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Q: 2. Decide whether each of the following sets are finite, denumerable or uncountable. Explain each of…

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Q: 1. (a) Consider the set S = {n(-1)" : n e N}. Verify the existence or otherwise of Min(S) and…

A: a) The given set is,S=n-1n:n∈ℕ

Q: (e) For the set W = {[1], [3], [13]} ≤ Z18 determine the set C = h¯¹(W). Then for the set C C Z12…

A: Modulo function

Q: (b) Consider the set Prove that inf A = 0. A = { n m² : n,m € Z¹, m≤n}.

A: Our objective is given below:

Q: (a) Show that for all n € Z, n ≥ 1, if A and B₁, B2, ... are any sets, then (c) n i=1 (b) Show that…

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Q: 10. S = {x² + 3r + 1, 2x² + x – 1, 4.x + 1} is: a. a linearly independent set of P2. b. a linearly…

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Q: 7. Use Green's Theorem to evaluate the integral f x²y dx - xy²dy where C: is the circle x² + y² = 4…

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Q: Which of the followings is a base of the set W = {[x 0 y] : x,y E R}:

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Q: 2. Let f: ZxZ-ZxZ be defined by f(m, n) = (3m+n,n²) a. Is fone-to-one? b. Is fonto?

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Q: (a) Consider the following true fact: For all n, m € Z \ {0}, if ged(n, m) Is the converse also…

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Q: 2. Use a table to express the values of each of these Boolean function. F(x, y, z) = x+yz

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Q: Q.No.3 a) Prove that if A and B are sets, then (An B)U (ANB') = A

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Q: Let: A = {n € Z|0< n < 3} B = {Vn|n €Nand 1 <n < 4}

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Q: (#4) Show that a set SCR" is closed if and only if bdry(S) CS.

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A: The sets are: (a) T=m∈Z | m=1+-1i for some integer i (b) U=r∈Z | 2≤r≤-2 (c) U=r∈Z | -2≤r≤2 To Do:…

Q: Let A and Bbe subsets of a universal set U and suppose n(U) = 400, n(A) = 200, n(B) = 160, and n(A N…

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Q: 6. Let z E C. Let C be set of points z satisfying Im(z) 0 and Im Prove that C = {ZEC : |Z| = 1}.…

A: Given that  Im1+z+z21-z+z2=0 We know that if Imz=0 then Im1z=0 Therefore  Im1-z+z21+z+z2=0 Finding…

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Q: 2. Let A = {0,1, 2, 4, 5, 7} and B = {2,3, 4, 5, 8} be subsets of the set %3D %3D U = {n € Z | 0 < n…

A: Given: A=0,1,2,4,5,7  &amp; B=2,3,4,5,8 is subset of U=n∈ℤ  0≤n≤9 i.e.U=0,1,2,3,4,5,6,7,8,9 To Find…

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Q: 2. (Problem 4.6) Let A, B be sets. (a) Prove that (b) Prove that P(ANB) P(A) P(B) P(AUB) CP(A) UP(B)…

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Q: Set X is convex if

A: Given that a set X is convex if  . . . .

Q: 36) Consider the set of points in the set C: C= {(x, y) |x, y E Z, x² + |y| < 2}. %3D Suppose that…

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Q: (b) Consider the following, more general, true fact: For all n, m € Z\ {0} and g € N, if ged(n, m)…

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Q: Let x e {1, 2} and y e {1, 4}, consider a predicate P(x, y) defined as P(x, y): 4 divides (x² + 3y).…

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Q: A = {n € Z | (3k e Z)(n B = {n E Z | (3j e Z)(n t A = B.

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A: Given that, In ℝ,ζυ, if A=1,3∩ℚ-ℤthen calculate the value of A.

Q: Suppose that A and B are two sets such that n[A] = 5, n[B] = 9, and n[AU B] = 13. If n[U] = 19, then…

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