HW7_Sol.pdf - DISCRETE MATHEMATICS – CH7 Homework7 7.1 10...

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DISCRETE MATHEMATICS – CH7 Homework7
7.1
10. If A = { w , x , y , z }, determine the number of relations on A that are (a) reflexive; (b)
symmetric; (c) reflexive and symmetric; (d) reflexive and contain ( x , y ); (e) symmetric
and contain ( x , y ); (f) antisymmetric; (g) antisymmetric and contain ( x , y ); (h)
symmetric and antisymmetric; and (i) reflexive, symmetric, and antisymmetric. (10 pts)
a. Reflexive 2
(𝑛
2
−𝑛)
= 2
(16−4)
= 2
12
b. Symmetric 2
𝑛
2
(𝑛
2
−𝑛)∗1/2
= 2
4
2
6
= 2
10
c. Reflexive and symmetric 2
(𝑛
2
−𝑛)∗1/2
= 2
6
d. Reflexive and contain (x,y) 2
12−1
= 2
11
e. Symmetric and contain (x,y) 2
𝑛
2
(𝑛
2
−𝑛)∗
1
2
−1
= 2
4
2
5
= 2
9
f. Anti-symmetric 2
𝑛
3
(𝑛
2
−𝑛)∗1/2
= 2
4
3
6
g. Anti-symmetric and contain (x,y) 2
𝑛
3
(𝑛
2
−𝑛)∗
1
2
−1
= 2
4
3
5
h. Symmetric and anti-symmetric {(1,1),(2,2),(3,3),(4,4)}
Relation either include or exclude each of these pairs, so 2
4
i. Reflexive, symmetric and anti-symmetric : only 1
{(1,1),(2,2),(3,3),(4,4)}
7.3
(10 pts)

2
7.4
12. Let A = { v , w , x , y , z }. Determine the number of relations on A that are (a) reflexive
and symmetric; (b) equivalence relations; (c) reflexive and symmetric but not transitive;