Let A and B be a pair of disjoint finite sets. Use induction to prove that if A ≈ m and B ≈ n, then A ∪ B ≈ m + n. Conclude that the union of two finite sets is finite.

Q: Find the sup and inf of the following set ∩(1 − 1/n, 2 + 1/n) where n ∈ N

A: The problem is asking us to find the supremum (least upper bound) and infimum (greatest lower bound)…

Q: Let S be the subset of the set of ordered pairs of integers defined recursively by Basis step: (0,0)…

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Q: Assume S is a recursively defined set of ordered pairs, defined by the following properties: (1,3)…

A: Given S is a recursively defined set of ordered pairs. We need to prove that 3a≤b whenever a,b∈S.…

Q: 1. Use the principle of mathematical induction to prove that 1 +r+r² + ³ + ... + pn = 11 pn+1 - r-1

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Q: Let n≥1, x be a real number, and x ≥ −1. Prove the following statement using mathematical induction.…

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Q: Let x, y and n denote integers. Use induction to prove that Vn 2 5: 3x,y 20 such that n = 2x + 3y

A: Strong induction: Let p(n) be a statement . Then (i) let p(n) is true for a base case p=m. (ii) If…

Q: Use mathematical induction to show that E"-, ar ark-1 a(1-r") for all integers n 2 1-r 1. (Assume r…

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Q: Use induction to show that A₁ U A₂ U U An|A₁|+|A₂|++|An, where A denotes the number of elements in…

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Q: Prove the statement using induction. (1+2+3+...+n} = 1³ +2³ +3³ +... n³ for every n € N.

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Q: Let xeN, yeN, and zEN. If y#z then x+y#x+z.

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Q: Let A and B be sets. Show that A∪(A∩B) =A using a traditional verbal proof, a series of equalities…

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Q: Exercise 1. The following lemma is important for the base case of the induction proof of the main…

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Q: Consider the reduction from 3-CNF-SAT to SUBSET-SUM. For the formula ϕ = (x1 ˅ ¬x3 ˅ x4) ˄ (¬x1 ˅ x3…

A: Given, Formula ϕ = (x1 ˅ ¬x3 ˅ x4) ˄ (¬x1 ˅ x3 ˅ x4) ˄ (¬x1 ˅ x2 ˅ ¬x4) and Clauses C, C1 = (x1 ˅…

Q: Prove T={ …,-56,-42,-28,-14,0,14,28,42,…} is a countable set

A: We have to prove the following -   T={ …,-56,-42,-28,-14,0,14,28,42,…} is a countable set.

Q: Use a proof by induction to prove the following: Let ao = 0, a₁ = 1, and a₁ = (ak-1 + ak-2 + 2) for…

A: We have to prove that 0<ak≤1 for all k using the proof by induction.

Q: Let k be a positive integer. Prove ÷VE. Note: do not use induction. j=1

A: The solution are next step

Q: V. Prove by induction that 1³ + 23 +3³ + +n³ = n²(n+1)² 4 for all positive integers.

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Q: Let A CN be a non-empty subset of natural numbers. Prove, using the principle of mathematical…

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Q: Write a proof for the General Multiplication Principle: Suppose that A , A2, ..., An are inite sets.…

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Q: Use mathmatical induction to prove that for all intergers n is greater than or equal to 1.…

A: For n = 1,       LHS = 4 [only 1 term]      RHS = 2.12 + 2.1 = 2.1 + 2.1 = 4 Hence, LHS = RHS.  So,…

Q: 6.) Prove by induction that 1/5 + 1/3/53 13 + *** ㅗ 402-1 n for every nz | 2n+1

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Set Theory

Every field in the present world involves a set. The theory of sets was developed by German mathematician Georg Cantor while working on the “problems of trigonometric series”.