Answer:
The "p if and only if q" statement is equivalent to "q if and only if p", then let's rewrite all the propositions in this way.
a) "If it is hot outside you buy an ice cream cone, and if you buy an ice cream cone it is hot outside."
We can define:
q = is hot outside
p = you buy an ice cream cone
Then we can rewrite this as:
"You buy an ice cream cone if and only if is hot outside"
b) "For you to win the contest it is necessary and sufficient that you have the only winning ticket."
We can define:
p = you winning the contest
q = have the winning ticket
Then we can rewrite (changing some words) this as:
"You will win the contest if and only if you have the winning ticke"
c) "You get promoted only if you have connections, and you have connections only if you get promoted."
We can define:
p = you get promoted
q = you have connections
We can rewrite the proposition as:
"You can get promoted if and only if you have connections"
d) " If you watch television your mind will decay, and conversely."
We can define:
p = you watch television
q = your mind will decay
We can rewrite it as:
"You can watch television if and only if your mind decays"
e) " The trains run late on exactly those days when I take it."
We can define:
p = The train runs late
q = I take it.
We can rewrite the proposition as:
"The train runs late if and only if I take it"
