In 9-33, determine whether the given relation is reflexive, symmetric, transitive, or none of these. Justify your answer.

Let X = { a , b , c } and P ( X ) be the power set of X. A relation L is defined on P ( X ) as follows: For every A,B A , B ∈ ( X ) , A L B ⇔ the number of elements in A is less than the number of elements in B.

To justify whether the given relation is reflexive, symmetric, transitive, or none of these.

Given information:

Let X = { a, b, c } and P(X) be the power set of X. A relation L Is defined on P(X) as follows: For all A, B ∈ P(X), A L B ⇔ the number of elements in A is less than the number of elements in B.

Calculation:

Consider the given set X={a,b,c}

Let us consider that A=P(X)

The relation L Is given by

L={B,C∈P(X)|B has less elements than C}

Reflexive:

The relation L Is reflective if (B,B)∈L for every element a∈A.

Since A=P(X), L Is reflexive if it contains (B,B) for all B∈P(X).

We note that E does not contain (B,B) because a set never has less elements than itself and thus E is not reflexive.

It is not reflexive

Symmetric:

The relation L on a set A is symmetric if (C,B)∈L whenever (B,C)∈L

Let us assume that (B,C)∈L, by definition of L :

B has less elements than C