Mathematics for Computer Science

“mcs” — 2017/3/3 — 11:21 — page 148 — #156
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Chapter 5
Induction
Problem 5.16.
Suppose P.n/ is a predicate on nonnegative numbers, and suppose
8k: P.k/ IMPLIES P.k C 2/: (5.17)
For P ’s that satisfy (5.17), some of the assertions below Can hold for some,
but not all, such P , other assertions Always hold no matter what the P may be,
and some Never hold for any such P . Indicate which case applies for each of the
assertions and briefly explain why.
(a) 8n 0: P.n/
(b) NOT.P.0// AND 8n 1: P.n/
(c) 8n 0: NOT.P.n//
(d) .8n 100: P.n// AND .8n > 100: NOT.P.n///
(e) .8n 100: NOT.P.n/// AND .8n > 100: P.n//
(f) P.0/ IMPLIES 8n: P.n C 2/
(g) Œ9n: P.2n/ IMPLIES 8n: P.2n C 2/
(h) P.1/ IMPLIES 8n: P.2n C 1/
(i) Œ9n: P.2n/ IMPLIES 8n: P.2n C 2/
(j) 9n: 9m > n: ŒP.2n/ AND NOT.P.2m//
(k) Œ9n: P.n/ IMPLIES 8n: 9m > n: P.m/
(l) NOT.P.0// IMPLIES 8n: NOT.P.2n//
Problem 5.17.
Consider the following sequence of predicates:
Q 1 .x 1 / WWD x 1
Q 2 .x 1 ; x 2 / WWD x 1 IMPLIES x 2
Q 3 .x 1 ; x 2 ; x 3 / WWD .x 1 IMPLIES x 2 / IMPLIES x 3
Q 4 .x 1 ; x 2 ; x 3 ; x 4 / WWD ..x 1 IMPLIES x 2 / IMPLIES x 3 / IMPLIES x 4
Q 5 .x 1 ; x 2 ; x 3 ; x 4 ; x 5 / WWD ...x 1 IMPLIES x 2 / IMPLIES x 3 / IMPLIES x 4 / IMPLIES x 5
: