00:01
In this problem, we will be looking at logical equivalences and recall that two statements are logically equivalent if in all possible cases, they have the same truth values.
00:12
So the first logical equivalence that we are looking at is this logical equivalence right here, where p and true is logically equivalent to just p.
00:22
And so our only variable in this logical equivalence, in fact, in all of the logical equivalences we will be looking at for this problem is p.
00:29
So in a truth table, we first write our variables, and p is our variable, which has two possibilities, true and false.
00:40
So then let's look at what happens when we have p and true.
00:46
So p when it's true, the statement p and true becomes true and true.
00:52
And recall that if you have an and statement, if both operands are true, it's true, otherwise it's false.
01:00
So if it's true and true, then it would be true.
01:04
Now, when p is false, the statement will become false and true, which again, if you have an n statement, both operands need to be true for the whole statement to be true.
01:13
So in that case, it would be false.
01:16
And it's pretty easy to see at the truth table right now that, in fact, in all possible cases, p and true has the same truth value as just p.
01:26
And therefore, it is logically equivalent to p.
01:31
The next logical equivalence we're looking at is if p or false is logically equivalent to p.
01:40
So let's look at what happens when we have p or false.
01:45
So when p is true, the statement becomes true or false.
01:48
And if you recall in an or statement or a disjunction, only one operand needs to be true or the whole statement to be true.
01:57
So when it's true or false, then the whole statement is.
02:02
Is true.
02:03
Now, on the other hand, if p is false and we have false or false, both operands are false, and so with an or operator, this evaluates to false.
02:19
And it's clear to see that in all of the possible cases, p or false has the same truth value as just p.
02:27
So this is logical equivalent, as we can see through the truth table...