Mathematics for Computer Science

“mcs” — 2017/3/3 — 11:21 — page 83 — #91
3.7. References 83
Problem 3.25.
Find a counter-model showing the following is not valid.
9x:P.x/ IMPLIES 8x:P.x/
(Just define your counter-model. You do not need to verify that it is correct.)
Problem 3.26.
Find a counter-model showing the following is not valid.
Œ9x: P.x/ AND 9x:Q.x/ IMPLIES 9x:ŒP.x/ AND Q.x/
(Just define your counter-model. You do not need to verify that it is correct.)
Problem 3.27.
Which of the following are valid? For those that are not valid, desribe a countermodel.
(a) 9x9y: P.x; y/ IMPLIES 9y9x: P.x; y/
(b) 8x9y: Q.x; y/ IMPLIES 9y8x: Q.x; y/
(c) 9x8y: R.x; y/ IMPLIES 8y9x: R.x; y/
(d) NOT.9x S.x// IFF 8x NOT.S.x//
Problem 3.28. (a) Verify that the propositional formula
is valid.
.P IMPLIES Q/ OR .Q IMPLIES P /
(b) The valid formula of part (a) leads to sound proof method: to prove that an implication
is true, just prove that its converse is false. 5 For example, from elementary
calculus we know that the assertion
If a function is continuous, then it is differentiable
is false. This allows us to reach at the correct conclusion that its converse,
5 This problem was stimulated by the discussion of the fallacy in [3].