Question

let A={1,2,3,4,5} and B={4,5,6} Let R is relation from A to B defined by R={(a,b):a+b ⊆ B} Then Find 1. R 2. Domain (R) 3. Range (R) 4.Inverse of R

Answer 1

1. Disregard my comment. Since R is said to be a relation from A to B, we take both a ∈ A and b ∈ A. Then the set R is the pairs (a, b) whose entries' sum belongs to B.

We have

• 4 = 1 + 3 = 2 + 2 = 3 + 1, so R contains (1, 3), (2, 2), and (3, 1)

• 5 = 1 + 4 = 2 + 3 = 3 + 2 = 4 + 1, so R contains (1, 4), (2, 3), (3, 2), and (4, 1)

• 6 = 1 + 5 = 2 + 4 = … = 5 + 1, so R contains (1, 5), (2, 4), …, (5, 1)

Then

R = {(1, 3), (1, 4), (1, 5), (2, 2), (2, 3), (2, 4),

…     (3, 1), (3, 2), (3, 3), (4, 1), (4, 2), (5, 1)}

2. The domain of R is the set of all values a in the pairs (a, b) belonging to R :

dom(R) = {1, 2, 3, 4, 5}

3. The range of R is the set of all values b in those same pairs:

range(R) = {1, 2, 3, 4, 5}

4. The inverse of R is the set with the same elements as R, but the entries are swapped:

inv(R) = {(3, 1), (4, 1), (5, 1), (2, 2), (3, 2), (4, 2),

…            (1, 3), (3, 2), (3, 3), (1, 4), (2, 4), (1, 5)}

In this case, we have R = inv(R).