Sets · Question 3

Let A = \( , ) R R : | - 1| 4 and | - 5| 6\ and B = \( , ) R R : 16( - 2)2 + 9( - 6)2 144\. Then

1. A B
2. B A
3. neither A B nor B A
4. A B = \(x, y) : -4 x 4, -1 y 11\

Answer

(2) B A

Solution

Set A: | - 1| 4 -3 5 and | - 5| 6 -1 11.

So A is the rectangle [-3, 5] [-1, 11].

Set B: 16( - 2)2 + 9( - 6)2 144 ( - 2)29 + ( - 6)216 1

This is the region inside the ellipse with centre (2, 6), semi-axes 3 (along ) and 4 (along ).

every point of B has -1 5 and 2 10

[-1, 5] [-3, 5] and [2, 10] [-1, 11], every point of B lies in A.

B A (A is not a subset of B, e.g. (-3, -1) A but B).

Also A B = A = \(x, y) : -3 x 5, -1 y 11\, so option (4) is false.

Answer: (2)