Question

Let R be the relation {(1, 2), (1, 3), (2, 3), (2, 4), (3, 1)}, and let S be the relation {(2, 1), (3, 1), (3, 2), (4, 2)}.

Find S ◦ R.

Answer 1

To find the composition of two relations S ◦ R, we need to match the second element of each pair in R to the first element of each pair in S and create a new pair. The resulting set of pairs is the composition S ◦ R = {(3, 2), (3, 3), (4, 1)}.

Step-by-step explanation:

To find the composition of two relations S ◦ R, we need to perform the composition operation by matching the second element of each pair in R to the first element of each pair in S.

Here is the step-by-step process:

1. Take the first element of each pair in R and search for a matching second element in S. If a match is found, create a new pair with the first element from S and the second element from R.
2. Repeat this process for each pair in R.
3. The resulting set of pairs is the composition of S ◦ R.

Applying this process to the given relations S and R gives us the composition S ◦ R = {(3, 2), (3, 3), (4, 1)}.

Answer 2

To compose S ▶ R, we compare pairs in R with pairs in S and create new pairs accordingly. After matching elements, we find that S ▶ R consists of the pairs {(1, 1), (1, 2), (2, 1), (2, 2)}.

Step-by-step explanation:

To find the composition of relations S ▶ R, you must take each ordered pair in R, find the second element of that pair in the pairs of S, and create a new pair from the first element of the R pair and the second element of the S pair. In other words, if you have (a, b) ∈ R and (b, c) ∈ S, then (a, c) will be in S ▶ R.

Using the given relations R = {(1, 2), (1, 3), (2, 3), (2, 4), (3, 1)} and S = {(2, 1), (3, 1), (3, 2), (4, 2)}, we compare the second element of each pair in R to the first element of each pair in S.

• (1, 2) in R matches (2, 1) in S, so (1, 1) is in S ▶ R.

• (1, 3) in R matches (3, 1) and (3, 2) in S, so (1, 1) and (1, 2) are in S ▶ R.
• (2, 3) in R matches (3, 1) and (3, 2) in S, so (2, 1) and (2, 2) are in S ▶ R.
• (2, 4) in R matches (4, 2) in S, so (2, 2) is in S ▶ R.
• (3, 1) in R does not match any pair in S because there is no pair starting with 1.

The composed relation S ▶ R is therefore {(1, 1), (1, 2), (2, 1), (2, 2)}.