## 1230.53 – Who is in Which Family?

Each of three men (Barber, Cutler, and Drake) is married to one of three women (Beth, Dorothy, or Louise) and each couple has a son (Allan, Henry, or Victor). You are to determine who is in each of these families using the following information.

1. Drake is neither Louise's husband nor Henry's father.
2. Beth is neither Cutler's wife nor Allan's mother.
3. If Allan's father is either Cutler or Drake, then Louise is Victor's mother.
4. If Louise is Cutler's wife, then Dorothy is not Allan's mother.

Solution

The families are:

Drake, Beth, Victor

Cutler, Dorothy, Henry

Barber, Louise, Allan

Solution of this problem is aided by the usual logic diagram. You will need three of them: One for husbands vs wives, one for husbands vs sons and one for wives vs sons. As you enter information in these diagrams, there is no difficulty entering given statements no. 1 and no. 2.

However, no. 3 and no. 4 present a problem: they are IF . . . THEN statements and we don't know which parts of them are true---only that the whole sentences are true. Attempts to solve the problem using just no. 1 and no. 2 are in vain. The information they give is obviously insufficient.

The key to the solution is to understand that a statement in the form IF p, THEN q is true in a variety of situations. Traditionally both p and q are thought of as true in such a statement, but an IF . . . THEN statement is also considered true if either p is false, or q is true (but NOT both!). In particular, both p and q may be false and in logic the statement is considered true. You may need to try several possibilities before finding the one that enables you to solve the problem.

Hint: Some of the possibilities give more information than others. In particular, the IF part of no. 3 gives a lot of information if it is false!