Question

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Problem 2. (12 points) Find a proposition with three variables p, q, and r that is false when p and r are true and q is false, and true otherwise.

Answer 1

The following proposition:

$\neg(p\land r) \lor q$

is false when p and r are true and q is false, and true otherwise. Notice that for above proposition to be false, both the terms of the disjunction have to be false i.e.

$\neg(p\land r)$ has to be false and   has to be false.

If q were true, above proposition will always be true.

Now, for $\neg(p \land r)$ to be false, $(p \land r)$ has to be true.

For $(p \land r)$ to be true, both p and r has to be true. Hence the above proposition is false only when both p and r are true, and q is false. And it will be true in all other cases. Please find the truth table of above proposition below:

p qr (p \ r) r(p /\ r) F F T F T F F T T T F T NOTE: T = TRUE, F = FALSE

From the truth table too, it can be clearly seen that the proposition is false only when p and r are true and q is false, and true for all other cases.

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