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questions 2-3) help

2019-10-16 (3).png

2019-10-16 (3).png

(b) (13) Use induction to prove that if n is any positive integer, and X, A1,A2, . . .An are
sets, then X\(A1UA2U...UAH)=(X\A1)fl(X\A2)n...n(X\An) (Note/hint: you may use results from the set-theory section of the course. You do
not need to use propositional logic as in question 1! As ever, Venn diagrams may
be good for intuition, but do not give real proofs.) Use induction to prove that 13 divides 42““ + 3“+2 for all n E N. Suppose that a = d - k + b, Where a, b, d, k are all integers. Prove that b is divisible
by d if and only if a is divisible by (1. Let x = abc be a three-digit number with digits a,b,c (so a, b, c 6 {0,1, 2, . . ..9})
Prove that a: is divisible by 3 if and only if a + b + c is divisible by 3. (Hint: if you’re stuck on the three-digit case, try proving the claim for two-digit
numbers 3,; = be to warm up.)

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