Answer to Question #182492 in Discrete Mathematics for EMMANUEL ARTHUR

Question

Let p denote He is rich and let q denote He is happy. Write each statement in symbolic form using p and q. Note

that He is poor and He is unhappy are equivalent to ¬p and ¬q, respectively.

(a) If he is rich, then he is unhappy. (c) It is necessary to be poor in order to be happy.

(b) He is neither rich nor happy. (d) To be poor is to be unhappy

Answer 1

I already said, that condition is little incomplete, so maybe it is required other form of answer. I have to guess in what form...

It could be:

a) $p \rightarrow \neg q$ - in logic natural language $\rightarrow$ means "than", so If he is rich, than he is unhappy (if he is rich and happy, than statement is false)

c) $\neg p \leftarrow q$ - if he is rich and happy, than statement is false (because it is necessary to be poor in order to be happy)

Other statements remain:

b) $\neg p \wedge \neg q$ -- Condition true only if he is is neither rich nor happy = he is poor and unhappy

d) $(\neg p \wedge \neg q) \vee (p \wedge q)$

Condition true if he is poor and unhappy or if he is rich and happy (To be poor is to be unhappy)

p.s. I think that from "To be poor is to be unhappy" it follows that "To be rich is to be happy" is also true.