Skill Sheet #2: Necessary and Sufficient Conditions

1. F is a sufficient condition for G if and only if the fact that something is F guarantees that it will also be G. Example: Driving legally is a sufficient condition for one's having a valid driver's license. The fact that a driver is driving legally guarantees that that driver has a valid driver's license.

2. F is a necessary condition for G if and only if the fact that something is not F guarantees that it will also not be G. Example: A necessary condition for driving legally is for the driver to obey traffic laws. The fact that a person is not obeying traffic laws guarantees that that person is not driving legally.

3. Take the phrase, “Being human is a necessary condition for being a woman, and being a woman is a sufficient condition for being human.” Put into plainer English, this phrase is saying: To be a woman, one must be human; conversely, if one is a woman, she must be human.

4. Definitions often take the form of listing several necessary conditions that are jointly sufficient. Example: Something is a doe if and only if it is a female deer.

Exercises

Consider the following conditions:

A: Part P is made with the proper material.

B: Part P is the part the customer has ordered.

C: Part P passed inspection.

D: Part P conforms to all of the customer's specifications.

Suppose for the purposes of this exercise that D is true if and only if A, B, and C are all true.

1. Is A a sufficient condition for D?

2. Is A a necessary condition for D?

3. Is the conjunction of A and C (that is, both A and C's being true) a sufficient condition for D?

4. Is the conjunction of A and C a necessary condition for D?

5. Consider the condition E: Part P must be made in less than an hour. In your opinion, is E a necessary condition for D, a sufficient condition for D, both necessary and sufficient for D, or neither necessary nor sufficient for D?

6. Is the conjunction of A, B, C and E sufficient for D?

7. Is this conjunction a necessary condition for D?

8. Is D a sufficient condition for B?

9. Is D a sufficient condition for the conjunction of C and E?