Quiz 7

Show all work in a neat and organized fashion. Clearly indicate your answers.

10 points possible.

1. (3 pts.) Write the contrapositive of the following statement.

For all positive real numbers p and q, if p=q, then Ö[pq]=(p+q)/2.

2. (4 pts.) A poker hand consists of 5 cards chosen from a standard deck of 52 cards. A hand is called ``two pairs'' provided it contains two sets of pairs of different rank (numerical value), with the fifth card having different rank from either of the pairs.

How many ``two pairs'' are there?

3. (3 pts.) There are exactly two possible equivalence relations on a two-element set: If A={1,2}, then R₁={(1,1), (2,2)} and R₂={(1,1), (1,2), (2,1), (2,2)} are the only equivalence realtions on A.

There are fifteen different equivalence relations on the set A={1,2,3,4}. You will be asked to list all of them.

Note: One of them is

R={(1,1), (1,2), (2,1), (2,2), (3,3), (3,4), (4,3), (4,4)}.

It would be painful to write out all of them like this. However, for short, we could just write the equivalence classes.

What are the equivalence classes for this R? Devise a convenient shorthand notation for your answer.

Now, using your shorthand notation, list all fifteen different equivalence relations on A={1,2,3,4}.