Discrte Math Proofs Completely prove number 2 for Category D CATEGORY A Prove th

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Discrte Math Proofs

Completely prove number 2 for Category D

CATEGORY A Prove that the product of any two consecutive integers has either the form 3k or 3k +2, k elementof Z. If n is an integer, then n^3 - n - 1 is always odd. CATEGORY B Prove that the number squareroot 3 is irrational. You may use the lemma "If 3 divides x^2, then 3 divides x" If a is a non-zero rational number and b is an irrational number, then 3a^2b/7 is an irrational number. CATEGORY C For all integers n greaterthanorequalto 20, 2^3n - 1 is divisible by 7 For all integers n greaterthanorequalto 1 and for r elementof R, r notequal 0, r notequal 1: Sigma^n_i = 0 r^i = r^n + 1 - 1/r - 1 CATEGORY D A sequence a_0, a_1, a_2, ellipsis defined by a_0 = 2, a_1 = 9 a_k = 5 a_k - 1 - 6 a_k - 2 for all integers k greaterthanorequalto 2 Prove that for all integers n greaterthanorequalto 0, a_n = 5 middot 3^n - 3 middot 2^n For all sets A, B, and C, (A - B) union (B - A) = (A union B) - (A intersection B)

Explanation / Answer

proof of

(B - A) U (A - B) = (A U B) - (A B)

let A and B be sets,

Defintion of set subtraction: .A - B .= .A B'

Axiom #1: .A A' = 0

Axiom #2: .A U 0 = A

Distributive Laws: .A U (B C) = (A U B) (A U C)
. . . . . . . . . . . . . .A (B U C) = (A B) U (A C)


The right side is: .(A U B) - (A B)

. . . . . . . . . . .= .(A U B) (A B)' . Def. of Subtraction

. . . . . . . . . . .= .(A U B) (A' U B') . DeMorgan's Law

. . . . . . . . . . .= .[(A U B) A'] U [(A U B) B'] . Distr.Law

. . . . . . . . . . .= .[(A A') U (B A')] U [(A B') U (B B')] . Distr.Law

. . . . . . . . . . .= .[0 U (B A')] U [(A B') U 0] . Axiom #1

. . . . . . . . . . .= .(B A') U (A B') . Axiom #2

. . . . . . . . . . .= .(B - A) U (A - B) . Def. of Subtraction

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