4. Let X be an ordered set, z E X, and SC X. Prove that x = sup S, if and only if Vy EX (yy).

Q: Prove that P(XY) = P(X)n P(Y) for all sets X and Y.

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Q: Let F= R and V = R*. Let W, = {(x1,x2,X3,X4) € R* : x1 – x2 + X3 – x4 = 0 and x1 +x2 = 0}, W2 =…

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Q: (c) Let X, Y and Z be sets. Prove that (X\Y)n (X \ Z) = X \ (YUZ).

A: We prove the given statement by using set theory.

Q: 1. Let A = {a, b, c}, B = {1,2,3}, C = {x, y, z, w}, D = {4,5, 6}. Let the functions f : A → B, g :…

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Q: Let u = (2, 3, 1), v = (1, 3, 0), and w = (2, -3, 3). Since (1/2)u -(2/3)v - (1/6)w = (0, 0, 0), can…

A: A set of vectors u1, u2, ……, un is said to be linearly independent if c1u1+c2u2+……+cnun=0 implies…

Q: Define relations S and R on the set A = {x, y, z, w} as follows S = {(y, x), (x, w), (w, z), (x,…

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Q: 1. Let X be a nonempty set of sets. Prove that P(UX) = U{P(A)| A e X} if and only if UX E X.

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Q: Let X={a, b, c, d, e} and let T={X, non zero, {a}, {a, d}, {a, e}, then (X, T) is neither Hausdorff…

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Q: Let C denote the set of all ordered pairs (a, b) with a,b & R. L.e., C:= {(a,b): a,b € R}. Define…

A: NOTE: According to guideline answer of first three related subpart can be given, for other please…

Q: Let x, y EZ. Prove that 3|xy if and only if 3|x x or 3 y.

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Q: Let W = (0, 1, 3}, X = {1, 3, 5), Y = (3, 4} and Z = {(1, 2), (3, 4), (1, 3),(2, 4). Compute (X x Y)…

A: The given problem is to find the Cartesian product of given sets and verify the Cartesian product…

Q: Let X = {1, (b) xm} and Y = {y₁, Yn} be finite sets. Determine the number of one to one functions…

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Q: Q. Let S = {(x, y), x² + 4y2 = 4} CR2, then S - {point} is homeomorphic to – y? = 4} a. {(x, y), 4x2…

A: Since you have asked multiple question, we will solve the first question for you. If you want any…

Q: Prove that if r>3 and y > 4, then ry > 11.

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Q: Suppose that A and B are sets and f : A –→ B is a function. Fix subsets X,Y C A. Prove that Imf(X) –…

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Q: 2. Let S = {0, 1, 2, 3, 4} and T = {0, 2, 4). (a) How many ordered pairs are in S × T? T × S? (b)…

A: 12. Given that,                    S=0, 1, 2, 3, 4  & T=0, 2, 4.

Q: 3. Show that is a closed subset of R². {(x, y) = R²: y = x²}

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Q: 2. Let A = {a, b, c}, B = {x, y, z}, C = {r, s,t}. Let f : A → B and g : B → C be defined by: f =…

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Q: 1. Use set identities to show that (A U B) n (B U A) = A where A and B are sets. Let A = {1,2, 3}…

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Q: -2 To 4 2. -5 -2 -2 %3D 17 - -4 -1

A: We have to find x and y

Q: Show that S = 2 is a linearly independent set in R³.

A: Given:     S = 121 , 011 , 201   To show that S is linearly independent set in R3

Q: Prove that if XY, then X UZ YUZ for all sets X, Y, and Z.

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Q: Let A = {0, 1, 2} and let B = {-2, 0, 1, 2}. %3D

A: Given,        A=0, 1, 2, and B=-2, 0, 1, 2

Q: Let x, y € Z. Prove that if 2 | (x² - y2) then 4 | (x² - y²)

A: We know that the sum of two odd numbers is an even number.Also, the difference between two odd…