Question

Let a = {4, 5, 6} and b = {6, 7, 8}, and let s be the "divides" relation from a to b. For every ordered pair (x, y) in a × b, x s y ⇔ x|y. Using set-roster notation, state explicitly which ordered pairs are in s and s^(-1). (Enter your answers as comma-separated lists of ordered pairs.)

a) {(4, 6), (5, 6), (6, 6)}

b) {(6, 4), (6, 5), (6, 6)}

c) {(4, 4), (5, 5), (6, 6)}

d) {(6, 4), (6, 5), (6, 8)}

Answer 1

The relation s consists of ordered pairs where the first element divides the second. The correct pairs for s are {(4, 6), (5, 6), (6, 6)}. The inverse s^(-1) contains the pair {(6, 6)}, as the elements in each pair from s are swapped and must still satisfy the divides relation.

Step-by-step explanation:

When dealing with the relation s, which is the 'divides' relation from set a to set b, we must identify the pairs where the first element divides the second element. The given sets are a = {4, 5, 6} and b = {6, 7, 8}.

To find which ordered pairs are in s, we look for instances where an element x in set a divides an element y in set b. After analyzing each possible pair, we can see that the only pairs that satisfy this condition are (4, 6), (5, 6), and (6, 6), because 4 divides 6, 5 divides 6, and 6 divides itself. Hence, the set s can be expressed in set-roster notation as {(4, 6), (5, 6), (6, 6)}.

Next, to find the inverse s^(-1), we take each pair in s and switch the elements, giving us pairs where the second element divides the first. However, only (6, 6) will remain since it is the only pair where both elements are equal and divide each other in both orders. thus, s^(-1) is {(6, 6)}.