Answer to Question #345450 in Discrete Mathematics for jessadel

Question

2. Prove that a relation R on a set A is symmetric if R^-1 = R.

3. Give an example of a relation that is reflexive but neither symmetric nor transitive.

4. Show that the relation ‘is perpendicular to’ over the set of all straight lines in the plane is symmetric but neither reflexive nor transitive.

5. Let S and T be sets with m and n element respectively. How many elements has S × T? How many relations are there in S × T?

6. If R and S are equivalence relations in the set X, prove that R ∩ S is an equivalence relation.

7. Show that the relation of congruence modulo m has m distinct equivalence classes.

8. Show that a partition of a set S deter- mines an equivalence relation in S.

9. Let S = {n: n ∈ N and n > 1}. If a, b ∈ S define a ~ b to mean that a and b have the same number of positive prime factors (distinct or identical). Show that ~ is an equivalence relation.

10. Prove that in the set N × N, the relation R defined by (a, b) R (c, d ) ⇔ ad = bc is an equivalence relation.

Answer 1

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