For each of the following “for all” statements, either disprove it with a specif
ID: 3420992 • Letter: F
Question
For each of the following “for all” statements, either disprove it with a specific counterexample, or prove it using one of the techniques we’ve studied. If you want to prove it, be sure to clearly indicate what you’re assuming, and what your goal is.
f) For all integers a,b,c, if a|b·c, then a|b or a|c.
g) For all integers a,b,c, if a|b or a|c, then a|bc.
h) For all integers greater than 3, 2n < n!
i) For all integers n > 0, 3 | (n – 1)(n+1)(n+3)
j) For all sets A,B, if there exists a set C such that then A = B.
k) For all sets A,B, if for all sets C, then A = B.
l) For all sets A,B, if there exists a set C such that A and , then A = B.
m) For all Boolean functions F,G,H of degree 2, if F(x,y) + H(x,y) = G(x,y) + H(x,y) for all x and y, then F(x,y) = G(x,y) for all x and y.
Explanation / Answer
k) For all sets A,B, if for all sets C, then A = B.
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