Relations and Functions

For any three sets A, B, C, prove that: A × (B ∪ C) = (A × B) ∪ (A × C)

If (x – y, x + y) = (0, 2), find x and y.

Find the domain of the function:f (x) = x² + 4x + 1 x² − 3x + 2 .

If A = {1,3} and B = {2,4}, find A × B. How many subsets will A × B have.

If A = {7, 11}, find A× A× A.

Let R be a relation from to N defined by R = {(a, b); a, b ∈ & N, and a = b²}. Are the following true.

(i) (a, b) ∈ R, for all a ∈ & N
(ii) (a, b) ∈ R ⇒ (b, a) ∈ R.
(iii) (a, b) ∈ R, (b, c) ∈ R ⇒ (a, c) ∈ R.

If A = {1,3,5}, B = {2,4}, C = {2,5,8}, find (A × B) ∩ (B × C).

Let f(x) = x² and g(x) = 3x + 2 be two real functions. Then, find:

(i) (f + g)(x) 

(ii) (f - g)(x) 

(iii) (fg)(x) 

(iv) (^f/_g)(x)

Find the domain and range of f(x) = ¹/_{√(x² - 16)}

Let A = {2, 3, 4,...,15}. Define a relation R from A to A by R = {(x, y); 3x - y = 0, where x, y ∈ A}. Write down its domain, codomain and range.

f(x) = x², find ^{f(1.1) - f(1)}/_{1.1 - 1}

Let f = {(1, 1), (2, 3), (0, -1), (-1, -3)} be a function from Z to Z defined by f(x) = ax + b, for some integers a, b. Determine a, b.

The cartesian product A × A has 9 elements among which are found {–1, 0} and {0, 1}. Find the set A and remaining elements of A × A.

The relation f is defined by

f(x) = { x², 0 ≤ x ≤ 3 
3x, 3 < x ≤ 10 }

The relation g is defined by

g(x) = { x², 0 ≤ x ≤ 2 
3x, 2 < x ≤ 10 }