# Various

1. Being an only child is (necessary, sufficient, or necessary and sufficient) condition for being a sibling? Explain?

2. Truth table for:

It is not the case that if the Lakers do not win the championship, then the coach will be fired. Therefore, if the Lakers do win the championship, then the coach will be fired.

Is the argument valid according to the truth tables?

3. An example of a conditional that is true, but its contapositive is false?

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#### Solution Preview

1. P is a necessary condition for Q if Q is true only when P is true.

P is a sufficient condition for Q if Q is always true when P is true.

The question is whether the relation between "X is an only child" and "X is a sibling" is either of these.

If we take P to be the proposition "x has at least one brother or one sister", then we can write "x is a sibling" as P and "x is an only child" as ~P. In other words, one is the *negation* of the other. In that case, can they *ever* have the same truth value? If not, can one be a necessary or sufficient condition for the other, as we've spelled this out above?

2. What you have to do here is make truth tables for each ...