Never understood this subject.

[quick fog]

Does it make sense to you how the table was set up?

p can either be true or false, q can either be true or false

These 4 are the only possibilities we can have

Both p and q true, p true but q false, now p false and q true or both false

No other possibilities. So our table needs these 4 rows

Does that make sense?

Now for each of these four cases, we give the truth value of them under some composition, like p -> q

If both p and q are true, then p implies q is defined as true

Nope, that'd be $\leftrightarrow$

[warped fjord]

Kepe

[quick fog]

aka "equivalence"

We define these compositions using truth tables

A -> B is defined like this:

You can often give them nice interpretations

[quick fog]

If the premise, A, is false, then that implies anything; A -> B will always be true.

If the premise, A, is true, then A -> B is only true if B is also true

Here is a common way to visualize this:

A: "It rains", B: "I take an umbrella with me". Now A -> B is "if it rains, I take an umbrella with me".

Now, we look at if I lied with that statement.

A and B both true: I take an umbrella with me and it rains. So I didn't lie. A -> B is true

A true, B false. It rains but I didn't take an umbrella with me. So I lied. A - >B is false.

A false, B true It doesn't rain and I still take an umbrella with me. Well, I didn't lie, I only said I will take an umbrella with me when it rains, since it didn't rain, I can do whatever. A -> B true

A false, B false. I don't take an umbrella with me and it doesn't rain. Well, I didn't lie. A -> B is true.

@sturdy trail

~ is the negation

It makes true out of false and false out of true

Ok, so the first three columns, up to q -> p make sense now?

Ok, now v means "or"

A v B is true if A is true or B is true (or both are true)

A v B is only false if A and B are both false

$\wedge$ means "and". \ $A \wedge B$ is true if $A$ is true $\emph{and}$ $B$ is true. So if e.g. $A$ is true and $B$ is false, it will be false.

Then we have everything defined. You should probably practice, e.g.

filling that table out

As for the question, $\equiv$ means "equivalent". Two statements are equivalent if they always have the same truth values.

So e.g. for the first thing in the problem, you compare the third column to the fourth column. If they have the same values everywhere, then they are equivalent

Yeah

So the first statement, $p \rightarrow q \equiv \sim p \rightarrow \sim q$ is false