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

Let A be the set of all strings of a’s and b’s of length 4. Define a relation R on A as follows: For every s , t ∈ A , s R t ⇔ s has the same first two characters as t.

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

Given information:

Let A be the set of all strings of a’s and b’s of length 4. Define a relation R on A as follows: For all s, t ∈ A, s R t ⇔ s has the same first two characters as t.

Calculation:

Reflexive:

Suppose s is any string in A. then sRs because s has the same first two characters as s.

Thus, reflexive

Symmetric:

Suppose s and t are any strings in A such that sRt.

By definition of R, s has the same first two characters as t.

Then t has the same first two characters as s, and so tRs