COMP SCI 242 Discrete Mathematics II Spring 2004

```COMP SCI 242 Discrete Mathematics II
Practice Exam
F. Baulieu
Each problem is worth 12 points except for problem 7 which is worth 16 points.
1) Give an example of a relation on { a, b, c, d, e } that is reflexive, symmetric, and not
antisymmetric. (Use any representation you like.)
2) On the set of all positive integers, define the relation  by x y iff the product xy is a perfect
square (square of an integer). Prove that  is an equivalence relation, and display two distinct
equivalence classes.
3) A Derangement of a sequence of distinct elements is an arrangement of the elements such
that no element is in its original position. Find a formula for the number of derangements of a
set of 5 distinct elements. You need not carry out the computation.
4) How many subsets of the set { 2, 5, 6, 7, 8, 12, 21 } contain at least one prime number? (You
need not carry out the computation.)
5) Prove that in any group of n people, there must be two who have the same number of friends
in the group. (Assume friendship is a symmetric relationship.) No one is considered to be their
own friend.
6) Let X be a set of n distinct objects. Find a formula for the number of reflexive symmetric
binary relations are there on X. (Hint: What will the matrix look like?)
7) Shown below is the relation matrix for a binary relation on the set {1, 2, 3, 4, 5, 6, 7 }
a) Display the digraph associated with this relation.
1
2
3
4
5
6
7
1
1
0
0
0
0
0
1
2
0
0
0
1
0
1
0
3
0
0
1
0
1
0
0
4
0
1
0
0
0
1
0
5
0
0
1
0
5
0
0
6
0
1
0
1
0
1
1
7
1
0
0
0
0
1
1
8) A balloon vendor sells bouquets of balloons. She has balloons in 5 different colors. How
many color combinations are possible for a bouquet of 10 balloons if it must contain at least one
blue balloon? (You need not carry out the computation.)
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