A truth table lists whether a statement is true or false.
If it's true, then it gets a T.
If it's false, then it gets an F.
Logic statements can only be true or false. There's no in between.
We can use variables to represent logical statements:
p: The art show was enjoyable.
q: The room was hot.
And we can use the following logical operators to refer to different logical scenarios:
~p means that p is false. (negation). In our example, ~p refers to “The art show was NOT enjoyable”.
^ means "and" (conjunction). For example, p ^ q refers to the statement “The art show was enjoyable AND the room was hot”.
v means "or" (disjunction) ). For example, p v q refers to the statement “The art show was enjoyable OR the room was hot”.
-> means "if then" or "implies". For example, q -> ~p refers to “if the room was hot, then the art show was not enjoyable”.
<-> means "if and only if" or "is equivalent to". For example, let:
p: The number is even.
q: The number is divisible by 2
Then we have: p <-> q
Let's look at a basic truth table. That table looks like this:
| p | ~p |
| T | F |
| F | T |
What this table is telling you is that:
if p is true, then ~p is false.
if p is false, then ~p is true.
p and ~p are negations of each other.
The AND statement truth table is as follows as it relates to p and q.
| p | q | (p ^ q) |
| T | T | T |
| T | F | F |
| F | T | F |
| F | F | F |
The AND statement requires p and q to both be TRUE in order for the AND statement to be true. Otherwise, the AND statement is false.
The OR statement truth table is as follows as it relates to p and q.
| p | q | (p v q) |
| T | T | T |
| T | F | T |
| F | T | T |
| F | F | F |
The OR statement requires p and q to both be FALSE in order for the OR statement to be false. Otherwise, the OR statement is true.
The IMPLIES statement truth table is as follows as it relates to p and q.
| p | q | (p -> q) |
| T | T | T |
| T | F | F |
| F | T | T |
| F | F | T |
If p is true and q is true, then the implied statement of p -> q is also true.
If p is true and q is false, then the implied statement of p -> q is false.
If p is false, then the implied statement of p -> q is always true, whether or not q is true or false.
As a matter of fact, p -> q doesn't apply when p is false. However, the implied statement itself, by the rules of logic, has to be true or false. There is no in between. Since the statement p -> q cannot be shown to be false when p is false, then the statement has to be true when p is false.
The rules for the IMPLIES statement are:
If p is true, then p -> q is true when q is true, and false when q is false.
If p is false, then p -> q does not appy but by rule of logic, is considered to be true whether or not q is true or false.
An IMPLIES statement is also called a "conditional" statement.
The EQUIVALENCY statement truth table is as follows as it relates to p and q.
| p | q | (p <-> q) |
| T | T | T |
| T | F | F |
| F | T | F |
| F | F | T |
p <-> q is another way of saying: "p is true if and only if q is true".
If both p and q are true, then the equivalency statement of p <-> q is true.
If both p and q are false, then the equivalency statement of p <-> q is still true.
In other words, if the truth table value of p agrees with the truth table value of q, then the equivalency statement of p <-> q is true. It doesn't matter if they're both true or both false.
An EQUIVALENCY statement is also called a "biconditional" statement.
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