Answer to Question #261544 in Discrete Mathematics for Amraj

Question

3. Let P(x), Q(x), and R(x) be the statements “x is a professor,” “x is ignorant,” and “x is vain,” respectively. Express each of these statements using quantifiers; logical connectives; and P(x), Q(x), and R(x), where the domain consists of all people

a) No professors are ignorant.

b) All ignorant people are vain.

c) No professors are vain.

d) Does (c) follow from (a) and (b)?

Answer 1

EXP1:

a) $\forall x:P(x)\to\neg Q(x)$

b) $\forall x:Q(x)\to R(x)$

c) $\forall x:P(x)\to\neg R(x)$

To check if (c) follows from (a) and (b) we can build a truth table. If there would be at least one row where statements (a) and (b) would be truth while statement (c) would be false, then (c) doesn't follow from (a) and (b)

As we see, on the set (1, 0, 1) we have truth for statements (a) and (b) and false for (c). So, (c) doen't follow from (a) and (b)

EXP.2:

a)

$\nexists x P(x)\iff Q(x)$

b)

$\forall x Q(x)\iff R(x)$

c)

$\nexists x P(x)\iff R(x)$

d) Answer: yes

since

a) : $\neg P\implies Q$

b) : $Q\implies R$

then:

c) : $\neg P\implies R$