# Given A = {1, 2, 3, 4, 5}, S = {(x, y) : x ∈ A, y ∈ A}.

Question:

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

Solution:

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)}.