Question

Write each of these propositions in the form “p if and only if q” in English.

a) If it is hot outside you buy an ice cream cone, and if you buy an ice cream cone it is hot outside.
b) For you to win the contest it is necessary and sufficient that you have the only winning ticket.
c) You get promoted only if you have connections, and you have connections only if you get promoted.
d) If you watch television your mind will decay, and conversely.

e) The trains run late on exactly those days when I take it.

Short answer

a) You buy an ice cream cone if and only if it is hot outside.

b) You win the contest if and only if you have only winning ticket.

c) You get promoted if and only if you have connections.

d) Your mind will decay if and only if you watch television.

e) The trains run late if and only if I take it.

Step-by-step solution

Definition of biconditional statement.

A bi-conditional statement is a combination of two conditional statements and its converse is written in the ‘if and only if’ form.

Bi-conditional statement in a)

If it is hot outside you buy an ice cream cone, and if you buy an ice cream cone it is hot outside.

The bi-conditional statement is,

You buy an ice cream cone if and only if it is hot outside.

Step 3: Bi-conditional statement in b)

For you to win the contest it is necessary and sufficient that you have the only winning ticket.

The bi-conditional statement is,

You win the contest if and only if you have only winning ticket.

Biconditional statement in c)

You get promoted only if you have connections, and you have connections only if you get promoted.

Bi-conditional statement is,

You get promoted if and only if you have connections.

Step 5: Biconditional statement in d)

If you watch television your mind will decay, and conversely.

Bi-conditional statement is,

Your mind will decay if and only if you watch television.

Step 6: Biconditional statement in e)

The trains run late on exactly those days when I take it.

Bi-conditional statement is,

The trains run late if and only if I take it.