Answer to Question #303547 in Discrete Mathematics for meher

Question

“If compound X is boiling, then its temperature must be at least 150◦C.” Assuming that this

statement is true, which of the following must also be true?

a. If the temperature of compound X is at least 150◦C, then compound X is boiling.

b. If the temperature of compound X is less than 150◦C, then compound X is not boiling.

c. Compound X will boil only if its temperature is at least 150◦C.

d. If compound X is not boiling, then its temperature is less than 150◦C.

e. A necessary condition for compound X to boil is that its temperature be at least 150◦C.

f. A sufficient condition for compound X to boil is that its temperature be at least 150◦C.

Answer 1

Let p be "compound X is boiling", q be "temperature must be at least 150◦C". Then statement “If compound X is boiling, then its temperature must be at least 150◦C.” can be written as $p\implies q$

(a) $q\implies p$ is not equivalent to $p\implies q$ , so it's false

(b) $\neg q\implies \neg p$ is equivalent to $p\implies q$ , so it's true

(с) $p\iff q$ is not equivalent to $p\implies q$ , so it's false

(d) $\neg p\implies \neg q$ is not equivalent to $p\implies q$ , so it's false

(e) $p\implies q$ is equivalent to $p\implies q$ , so it's true

(f) $q\implies p$ is not equivalent to $p\implies q$ , so it's false