Q2: Let R be the parent relation on the set of all people (see Example 21 in section 9.1 of the book). When is an ordered pair in the relation R^3? SUGGESTED TEXT: · Keneth H. Rosen. Discrete Mathematics and its Applications. 7th edition.

Solution:

Given from this example: Let R be the relation on the set of all people such that (a, b) ∈ R if person a is a parent of person b. Then (a, c) ∈ R ◦R if and only if there is a person b such that (a, b) ∈ R and (b, c) ∈ R, that is, if and only if there is a person b such that a is a parent of b and b is a parent of c. In other words, (a, c) ∈ R ◦R if and only if a is a grandparent of c.

Now, $R^3=R^2\cdot R=(R\cdot R)\cdot R$

Say, $(a,c)\in R\cdot R$ , then a is grandparent of c.

Now, $(a,d)\in (R\cdot R)\cdot R$ , then a is great grandparent of d.

So, $(a,d)\in R^3$ if and only if a is a great grandparent of d.