Exercises 7.1:

Work:

Let A = {a1,a2,…an}
If R is a relation on A , R AxA and if R is antisymmetric also:

If (ai,aj)R and (aj,ai)R then a i = a j

Then if (ai,aj)R with i j then (aj,ai) R
So we can construct R where Ris maximum taking:
R = {(ai,ai) , 1 i n} U {(ai,aj) , 1 i < j n}
R= n + n2 = n+n(n-1)/2 = n[1+(n-1)/2] = n(n+1)/2
Answer:
Maximum value for R= n(n+1)/2 and it´s only 1 antisymmetric relation with this property (the one constructed above)

Exercises 7.2:

Work:
We know that (a,a)R for all aA
Since (a,a) R and (a,a) R if we apply definition of R2
(a,a) R2 for all aA then R2 is a reflexive relation on the set A

Work:
Each entry is 0 or 1 and we must count how many matrices 6x6 satisfy A = A tr
The number of matrices that A = Atr is equal to the number of matrices that i construct selecting the entries in the diagonal and above.
1+2+3+4+5+6 = 21
Answer: 21

Exercise 7.3:

a)
Relation matrix is M=1 1 1 1 10 1 0 1 10 0 1 1 10 0 0 1 10 0 0 0 1
b)
I will do it by hand
c)
Work:
Using the Fig 7.23
For k = 1 e is the only vertex that has no edges starting at it
Then we remove e
Result: e
For k = 2 d is the only vertex that has no edges starting at it
Then we remove d
Result: d < e
For k = 3 b and c are the only vertices that has no edges starting at it we select c (it can be b too)
Then we remove c
Result: c<d<e
For k = 4 b is the only vertex that has no edges starting at it
Then we remove b
Result: b<c<d<e
For k = 5 (k =n) a is the only vertex that has no edges starting at it
Then we remove a
Result: a<b<c<d < e
Answer: a < b < c < d < e
Exercise 7.4:

Work:

a)

Since A1UA2UA3 = A and AiAj = (i,j I={1,2,3}, ij} , yes {A1,A2,A3} it is a partition of A

Answer: Yes

b)

Since A1A2 ={d} then