Given A = {1, 2, 3, 4, 5}, S = {(x, y) : x ∈ A, y ∈ A}. Find the ordered pairs which satisfy the conditions given below:
(i) x + y = 5
(ii) x + y < 5
(iii) x + y > 8
According to the question, A = {1, 2, 3, 4, 5}, S = {(x, y) : x ∈A, y ∈A}
(i) x + y = 5
So, we find the ordered pair such that x + y = 5, where x and y belongs to set A = {1, 2, 3, 4, 5},
1 + 1 = 2≠5
1 + 2 = 3≠5
1 + 3 = 4≠5
1 + 4 = 5⇒ the ordered pair is (1, 4)
1 + 5 = 6≠5
2 + 1 = 3≠5
2 + 2 = 4≠5
2 + 3 = 5⇒ the ordered pair is (2, 3)
2 + 4 = 6≠5
2 + 5 = 7≠5
3 + 1 = 4≠5
3 + 2 = 5⇒ the ordered pair is (3, 2)
3 + 3 = 6≠5
3 + 4 = 7≠5
3 + 5 = 8≠5
4 + 1 = 5⇒ the ordered pair is (4, 1)
4 + 2 = 6≠5
4 + 3 = 7≠5
4 + 4 = 8≠5
4 + 5 = 9≠5
5 + 1 = 6≠5
5 + 2 = 7≠5
5 + 3 = 8≠5
5 + 4 = 9≠5
5 + 5 = 10≠5
Therefore, the set of ordered pairs satisfying x + y = 5 = {(1,4), (2,3), (3,2), (4,1)}.
(ii) x + y < 5
So, we find the ordered pair such that x + y<5, where x and y belongs to set A = {1, 2, 3, 4, 5}
1 + 1 = 2<5 ⇒ the ordered pairs is (1, 1)
1 + 2 = 3<5 ⇒ the ordered pairs is (1, 2)
1 + 3 = 4<5 ⇒ the ordered pairs is (1, 3)
1 + 4 = 5
1 + 5 = 6>5
2 + 1 = 3<5 ⇒ the ordered pairs is (2, 1)
2 + 2 = 4<5 ⇒ the ordered pairs is (2, 2)
2 + 3 = 5
2 + 4 = 6>5
2 + 5 = 7>5
3 + 1 = 4<5 ⇒ the ordered pairs is (3, 1)
3 + 2 = 5
3 + 3 = 6>5
3 + 4 = 7>5
3 + 5 = 8>5
4 + 1 = 5
4 + 2 = 6>5
4 + 3 = 7>5
4 + 4 = 8>5
4 + 5 = 9>5
5 + 1 = 6>5
5 + 2 = 7>5
5 + 3 = 8>5
5 + 4 = 9>5
5 + 5 = 10>5
Therefore, the set of ordered pairs satisfying x + y< 5 = {(1,1), (1,2), (1,3), (2, 1), (2,2), (3,1)}.
(iii) x + y > 8
So, we find the ordered pair such that x + y>8, where x and y belongs to set A = {1, 2, 3, 4, 5}
1 + 1 = 2<8
1 + 2 = 3<8
1 + 3 = 4<8
1 + 4 = 5<8
1 + 5 = 6<8
2 + 1 = 3<8
2 + 2 = 4<8
2 + 3 = 5<8
2 + 4 = 6<8
2 + 5 = 7<8
3 + 1 = 4<8
3 + 2 = 5<8
3 + 3 = 6<8
3 + 4 = 7<8
3 + 5 = 8
4 + 1 = <8
4 + 2 = 6<8
4 + 3 = 7<8
4 + 4 = 8
4 + 5 = 9>8, so one of the ordered pairs is (4, 5)
5 + 1 = 6<8
5 + 2 = 7<8
5 + 3 = 8
5 + 4 = 9>8, so one of the ordered pairs is (5, 4)
5 + 5 = 10>8, so one of the ordered pairs is (5, 5)
Therefore, the set of ordered pairs satisfying x + y > 8 = {(4, 5), (5, 4), (5,5)}.