Let A = {1, 2}, B = {1, 2, 3, 4}, C = {5, 6} and D = {5, 6, 7, 8}. Verify that- A x (B ∩C) = (A x B) ∩(A x C)
- A x C is a subset of B x D
- A x (B ∩C) = (A x B) ∩(A x C)
- A x C is a subset of B x D
Solution and Explanation
(i) To verify: A x (B∩C) = (A x B)∩(A x C)
We have B ∩C = {1, 2, 3, 4}∩{5, 6} = Φ
∴L.H.S. = A x (B∩C) = A x Φ = Φ
A x B = {(1, 1), (1, 2), (1, 3), (1, 4), (2, 1), (2, 2), (2, 3), (2, 4)}
A x C = {(1, 5), (1, 6), (2, 5), (2, 6)}
∴ R.H.S. = (A x B) ∩(A x C) = Φ
∴L.H.S. = R.H.S
Hence, A x (B ∩C) = (A x B) ∩(A x C)
(ii) To verify: A x C is a subset of B x D
A x C = {(1, 5), (1, 6), (2, 5), (2, 6)}
B x D = {(1, 5), (1, 6), (1, 7), (1, 8), (2, 5), (2, 6), (2, 7), (2, 8), (3, 5), (3, 6), (3, 7), (3, 8), (4, 5), (4, 6), (4, 7), (4, 8)}
We can observe that all the elements of set A x C are the elements of set B x D.
Therefore, A x C is a subset of B x D.
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Concepts Used:
Relations and functions
A relation R from a non-empty set B is a subset of the cartesian product A × B. The subset is derived by describing a relationship between the first element and the second element of the ordered pairs in A × B.
A relation f from a set A to a set B is said to be a function if every element of set A has one and only one image in set B. In other words, no two distinct elements of B have the same pre-image.
Representation of Relation and Function
Relations and functions can be represented in different forms such as arrow representation, algebraic form, set-builder form, graphically, roster form, and tabular form. Define a function f: A = {1, 2, 3} → B = {1, 4, 9} such that f(1) = 1, f(2) = 4, f(3) = 9. Now, represent this function in different forms.
- Set-builder form - {(x, y): f(x) = y2, x ∈ A, y ∈ B}
- Roster form - {(1, 1), (2, 4), (3, 9)}
- Arrow Representation
