3. Prove that if Z₁ Z₂ Z3 = 0, then at least one of the three factors is zero. 4. Show that a set S is unbounded if and only if every neighborhood of the point at infinity contains at least one point in S.

Q: Let P1, P2, P3, .. , P78 be some coplanar points with k lines on the same plane. If exactly two line…

A: Given,  P1, P2,P3,...,P78 be some coplanar points with k lines on the same plane. If exactly…

Q: Let a and b be integers, at least one of them not 0. Prove that an integer e can be expressed as a…

A: Let a and b be intgers ateast one of them not 0. Our Aim is to prove that an integer e can be…

Q: 2. Let x be an element of a convex set S. Assume that x1 = x + €ip E S and x2 = x – €2P E S, where p…

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Q: Exercise. Consider the set R. For every x = (x1, x2) E R? and y = (y1, y2) E R², write Pi(x, y) =| x…

A: (a) Here, ρ1x, y=x1-y1+x1-y2   (1) show that ρ1x, y≥0 : We know that, x≥0,  ∀x Therefore, x1-y1≥0,…

Q: Describe geometrically the set of points z satisfying the following condition: |z +1– i|< |z – 1+…

A: This question is related to complex numbers.

Q: (a) Is the set S = {1 + x + x2, 1 − x3,  x + x2 + x3,  x3} ⊂ P3 linearly independent? If so, prove…

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Q: Assume that a and b are real numbers such that a <b. Prove that the interval (a,b) is an open set.…

A: Given that a and b are real numbers and a&lt;b. Prove that (a,b) is an open set

Q: Prove that the set A = {x ∈ R | 3x + 2 ≤ 0} has a maximum

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Q: Let S = {[k 1 0],[1 k 1],[0 1 k]}. Deter- mine all the real values of k so that S is a linearly…

A: Given that, S = {[k    1    1], [1     k    1], [0    1    k]} If this set of vectors is linearly…

Q: d) Give a recursive definition of the sequence {a„}, n = 1, 2, 3... %3D n if a,= 2 + 1; and a15 e)…

A: d)  Here, given that an=2+1=3 So, for all n, an=3 which is the recursive definition.

Q: 2. Prove that is real if and only if z = z. Z Y

A: Since you have uploaded multiple questions, according to our guidelines we have solved first…

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: 3. Consider the following statement and ‘proof.’ Statement. Let x, y ∈ Z. If x3 − y2 is odd, then…

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Q: 3. Let R denote the set of real numbers. For each of the following subsets of Rx R, determine…

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Q: prove for every two sets A and B that A=(A-B)U(A n B)

A: To prove: We want to prove  for every two sets A and B that A=A-B∪A∩B

Q: 5. Let V₁, V2, V3, and v4 be such that {V1, V2, V3} is linearly dependent and {v2, V3, V4} is…

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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: 1. Find all regular points of the set : {x € Rª | 2(x1+x2+x3)³ +3(x1+x2+x3)² = 1, x²+x²+x²+x4 = 1}.

A: The given set is: x∈R4|2(x1+x2+x3)3+3(x1+x2+x3)3=1,x12+x22+x32+x4=1

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: Prove that if C is a point of −→AB other than A, then −→AC = −→AB.

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Q: Determine which of these relations are reflexive. The variables x, y, x', y' represent integers. A.…

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Q: Direction: Define subsets of Za, as follows: A = {a | a E Za and a 5} A = {a | a E Za and 3| a} 1.…

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Q: 5. Consider the following statement: For all sets A, B, and C, then AU (BNC) = (AUB) nC. Either…

A: Let value of set A,set B and set C are respectively. A={1,2,3,6,7} B={3,7,8,9} C={2,3,6,9}

Q: Express the characteristics of the following sets of points, in terms of: closed or open, connected…

A: The set of points {z∈C: |arg (z)|&lt;(π/2)} is:   Closed: The set includes all complex numbers…

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: Exercise 3.3.16. Prove or find a counterexample to the following statement. Let A, B and C be sets.…

A: Solution of the question is given below.

Q: Let (pos S) be the set of all positive combinations of the points of S. Which of the following…

A: We need to find out which statements are correct. To do that, we shall independently analyze both…

Q: In Problems 19-28, describe in words the set of all points (x, y, z) that satisfy the given…

A: 24 We have to describe the set of all points x,y,z that satisfy the statement x=y and z=0.…

Q: Ex 12.8. Let S be a set of q + 2 points in a projective plane of order q. Prove that every line,…

A: Given: In this question, the given details are, Let S be a set of q+2 points in a projective plane…

Q: A set L cR is convex if:

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Q: Exercise 3.3.15. Prove or find a counterexample to the following statement. Let A, B and C be sets.…

A: We have a statement  Let A, B and C be sets. Then (A - B) U C = (A U B U C) - (A ∩ B).

Q: Let B be a subset of R¹. Which of the following statements are correct? If B is a base, then |B| =…

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