Few propositional logic questions

[Panphobia]

Homework Statement

Are these propositions, if so are they true or no?

a. \sqrt{n} = 2

b. Consider an integer n: \sqrt{n} = 2 and n = 4

c. Consider an integer n: if \sqrt{n} = 2 then n = 4

Here is another question.

Translate the following into a propositional expression involving two propositions p and q.

d. Philip gets caught whenever he cheats.

The Attempt at a Solution

a. I would this this is a proposition and it is false because we don't know what n is defined as.
b. Proposition and false because we don't know what n is defined as.
c. Proposition and true because FALSE → FALSE = TRUE

d. p = Philip gets caught
q = he cheats
I am not sure if this is an, if and only if, or just an if then.

 

[U.Renko]

A proposition is something you can rephrase as "IF something THEN something else"

sometimes is not very explicit, sometimes is quite obvious

however be careful with saying FALSE just because so and so is not defined.
for example: in b the statement itself is defining n:
n is the number whose square roott is 2

 

[Panphobia]

Yes but sometimes wasn't in any of these. I am thinking it is, if p then q, more because Philip getting caught is not contingent on him cheating, he could be caught doing something else.

 

[U.Renko]

he could be caught doing something else.

Sure.

but whenever he cheats he undoubtly gets caught.

do not confuse "if he cheats then he gets caught" (he's bad at hiding it or something)
with
"if he gets caught then he was cheating" (he can get away with anything except cheating)

these two are completely different statements

 

[Panphobia]

So the answer is, if he cheats then he gets caught

 

[U.Renko]

yup!

 

[Panphobia]

Another one is pretty confusing also, "Getting elected follows from knowing the right people". So if p = getting elected and q = knowing the right people, so if p then q works, but p iff q also works, if q then p doesn't work. Is it p iff q or if p then q?

 

[U.Renko]

this one is phrased confusingly indeed

but remember:
in order to "iff" to work:

BOTH "if p then q" AND "if q then p" have to work