By showing that each side is a subset of the other side, prove that A △ B = (A ∪ B) − (A ∩ B)

Q: Prove that if alb and alc then a (bx + cy) for all integers x, y.

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Q: Given A,B and C required to prove that A\(B\C)=(A\B)U(A∩B∩C)

A: Given  A, B, C are any three sets  We have to prove  A\(B\C)=(A\B)U(A∩B∩C)    Let X be unversal set…

Q: Use a) and b) to find A-1

A: Let's find inverse of A from given results a and b.

Q: 1. Suppose we would like to define a function f : Q→ Z by the formula f(r) = a +b, where a, b € Z…

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Q: If A={a,b,c,d,e,f} and B={k,l,m,n}, then find the Cartesian product ? × ? of A and B

A: Set of Each entry in 2 tuple in the below table is Cartesian product of A and B

Q: Show that If (A ∩ B = ø) then P (A ∪ B) = P (A) + P (B)

A: Given that A ∩ B = ø This means P(A ∩ B ) = 0

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: Show that if A ⊂ B, then C – B ⊂ C – A.

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Q: Is the following true or false? Z∩N=Z

A: Z is the set of all integers, i.e.  Z={ ..., -3, -2, -1, 0, 1, 2, 3, ...} And N is the set of all…

Q: Prove: For all sets A, B and C, (AUB) - (CA) = AU (B-C).

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Q: z = (a + bi)4n+ (b + ai) 4n; a ∈ ℝ ; b ∈ ℝ; n ∈ ℕ. Show that z is a real number.

A: Properties of complex number : If z is real number than z=z¯ . Suppose z=x+i y than z¯=x- iy .…

Q: etermine whether the set of x-2x²,2+5x-x²x+x²} is linerly independent in P

A: First we use definition of linearly independent Set and we check given vectors are linearly…

Q: Let A = {1, 2}, B = {1, 2, 3, 4}, C = {5, 6} and D = {5, 6, 7, 8}. Verify that(i) A × (B ∩ C) = (A ×…

A: i) Answer(i) A × (B ∩ C) = (A × B) ∩ (A × C) (proved)

Q: a) Show that the set S = {(1, 2, 3), (0, 1, 2), (– 2, 0, 1)} spans R³. b) Determine whether the set…

A: See the attachment.

Q: Let A and B be the multisets {3 ⋅ a, 2 ⋅ b, 1 ⋅ c} and {2 ⋅ a, 3 ⋅ b, 4 ⋅ d}, respectively. Finda) A…

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Q: Let there be set of a, b, c. Prove A×(B∩C) = (A∩B)×(B∩C)

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Q: Let a, b, q, r E Z and assume that a and b are not both zero and b and r are not both zero. Show…

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Q: Exercise 2. Let A, B C R. (a) Prove that AUB = AUB. (b) Prove that ANBCAN B. (c) Find sets A and B…

A: Solution:-

Q: d) Suppose X = {1,2, 3} and Y = {5,6}. The number of elements of the set P(X× Y) =

A: Elements in X = 3. Elements in Y = 2 Elements in X × Y = 3 × 2 = 6

Q: df Catx) is expandad

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Q: Let 7 -7 2 -2 -5 %3D - 3 -2 9. -8 a) Find the dimension of the row space ofA. b) Find the dimension…

A: The given matrix,A=117-72-2-565-3-25409-8

Q: a) Let A = {4, 7, 8} and B = {−1, 5}, Find and compare sup(A+B) , inf(A+B), sup(A)+sup(B),sup A -…

A: See the attachment

Q: Is a R a maximal deal

A: No, it is not necessarily true that aR is a maximal ideal in a commutative ring R with unity. In…

Q: Let the sets A, each bo countable. (a) Show that the countable union U₁A; is countable (6) Show that…

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Q: Determine if v is in the set spanned by the columns

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Q: For the four pairs of sets below, determine whether or not they are equipollent. i.) {2, 5, 10} and…

A: We have to find the equipollnt sets from the given pair of four sets.

Q: Does A − (B − C) = (A − B) ∪ C hold for all sets A, B, and C? Prove or give a counterexample for…

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Q: Let A and B be two sets. We denote A- B = { x | x∊A and xÏB}. Prove that A∪B=(A−B)∪(A∩B)∪(B−A).

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Q: Let V be all elements of R2 that have the form (a, a2) where a is a real number. Show that (3,9) is…

A: Let V be all elements of R2 that have the form (a, a2) where a is a real number. We have show that…

Q: If a1, az, az,.-ans. be the A.P. and S, = a-1m+1+a-1m+2+a-1n+3 +.--+ am %3D Prove that :…

A: In this question we have to proof S1, S2, S3,.... are in Arithmetic Progression.

Q: Let A, B, C be sets. Prove that if C ⊆A, then A −(B −C) = (A −B) ∪C. Analogously, for any real…

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Q: Is it true that (A×A)∖(B×B)=(A∖B)×(A∖B) for any two sets A, B? If it is true, prove it. Otherwise,…

A: Cartesian product and identity

Q: let A, B ⊂ R, and suppose A and B are closed. Prove that A ∩B is closed. Hint: You might try showing…

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Q: If A ⊂ B, can A and B be independent under any circumstances? Explain

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Q: Let A,B,C,D be sets. Prove that (A∩B)×(C∩D)=(A×C)∩(B×D)

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Q: Let A={1,2,3,4,5} and B={0,2,3}. Find 1)A∪B 2) A ∩ B 3) A - B 4) A ⊕ B

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