> 11 1
> 10 0
> 01 1
> 00 1
>Can someone explain the "IF" table?
This is a very common question. To some degree, as Marchal suggests,
you can think of it as a definition.
But to motivate it, suppose Sally asks her father if he will take her to
the zoo. He tells her, "if you are a good little girl, I will take you."
Now let us consider whether her father is telling the truth. Suppose
Sally is good and he does take her. In this case his statement is true.
Suppose Sally is good but he does not take her. Clearly in this case
his statement was not true.
Now suppose Sally was bad, and he does not take her to the zoo. I think
we will agree that her father's statement was true in this case.
The final case is that Sally was bad, but her father takes her to the
zoo anyway. Technically his statement was still true. He said he would
take her to the zoo if she was good, but he didn't say what he'd do if
she was bad. So this outcome is consistent with his statement and it
is considered true.
This leads to the truth table above.
A similar relationship is expressed as "if and only if", or "equivalence".
Here her father says, "I will take you to the zoo if and only if you
are good." This means that if she is good, he will take her, but if
she is bad, he will not take her. The truth table is:
And we see that C is true when A and B have the same value.
Sometimes when we use "if" in common language we really mean "if and
only if", other times we mean the weaker "if" used in symbolic logic.
For the weaker "if", the statement is automatically true any time the
antecedant (the A above) is false.