Page 200 - HOW TO PROVE IT: A Structured Approach, Second Edition

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186 Relations
−1
must prove that R = R , and this is done by proving both R ⊆ R −1 and
R −1 ⊆ R. Each of these goals is proven by taking an arbitrary element of the
first set and showing that it is in the second set. In the ← half we must prove
that R is symmetric, which means ∀x ∈ A∀y ∈ A(xRy → yRx). We use the
obvious strategy of letting x and y be arbitrary elements of A, assuming xRy,
and proving yRx.

Exercises

∗
1. Let L ={a, b, c, d, e} and W ={bad, bed, cab}. Let R ={(l, w) ∈
L × W | the letter l occurs in the word w}. Draw a diagram (like the
one in Figure 1) of R.
2. Let A = {cat, dog, bird, rat}, and let R ={(x, y) ∈ A × A | there is at least
one letter that occurs in both of the words x and y}. Draw a directed graph
(like the one in Figure 3) for the relation R.Is R reflexive? symmetric?
transitive?
∗
3. Let A ={1, 2, 3, 4}. Draw a directed graph for the identity relation on
A, i A .
4. List the ordered pairs in the relations represented by the following di-
rected graphs. Determine whether each relation is reflexive, symmetric,
or transitive.