Math 187 Review Problems III
1. There are forty couples. Ten of the couples are green, ten are purple, ten are blue, and ten
are red.
a. In how many ways can these people stand in a line?
b. In how many ways can these people stand in a circle?
c. In how many ways can these people stand in a line if they must alternate in
gender?
d. In how many ways can these people stand in a circle if they must alternate in
gender?
e. Suppose we select twelve of these people. In how many ways can we select two
sets of six people each, if the two sets of people must be of a different color?
f. Suppose we select twelve of these people. We select six people of one color and
three people each of two different colors. So we have three different colors
represented in our selection of twelve people. In how many ways can this
selection of twelve people be made?
2. Write the following sentences using quantifier notation (i.e., use the symbols and/or ).
Do not attempt to prove these statements.
a. There is an integer that, when multiplied by any integer, always gives the result 0.
b. No matter what integer you choose, there is always another integer that is larger.
c. Everybody loves somebody sometime.
3. Now write the negation of the three statements above.
4. Let
and be sets. Prove the following DeMorgan’s Laws.
(
) (
) (
)
a.
(
) (
) (
)
b.
| | || |.
5. Let and be finite sets. Prove: |
| | | | | if, and only if,
6. Let and be finite sets. Prove: |
.
7. Let
and be sets. Prove or disprove:
| | | | | | |.
If
, then |
*( ) ( ) ( )+. Is a function? Why or why not?
8. Let
9. Let
by ( )
. Prove: is one-to-one and onto .
(
) (
).
10. Let and be finite sets. Prove:
11. Let be the set of even integers. Let be the set of odd integers. Prove:
by ( )
is a bijection.
12. Let be a function. Prove:
is a function if, and only if, is one-to-one.
13. Let and be sets. Let
. Prove:
if, and only if, is one-to-one
and onto .
14. Let and be finite sets and let
. Prove that any two of the following
statements being true implies the third (note that this requires three separate proofs).
a.
is one-to-one.
b.
is onto.
c. | | | |.
15. Let
with
. Recall ( )
. Prove:
(
)
( )
(
)
(
)