Relations and Functions

Exercise 1

Q – 1 – If find the values of x and y.

Solution:

Given:

This means;

Q - 2 - If the set A has 3 elements and the set B = {3, 4, 5}, then find the number of elements in (A x B)

Solution:

Here givenof elements in set A = 3]

n(B) = 3 [ number of elements in set B = 3]

Therefore,

n(A X B) = n(A) . n(B)

= 3 x 3

= 9

Hence, the number of elements in (A X B) = 9Hence, the number of elements in (A X B) = 9

Q – 3 – If G = {7, 8}, H = {5, 4, 2}, find G x H and H x G.

Solution:

G = {7, 8} and H = {5, 4, 2}

G X H = {7, 8} X {5, 4, 2}

            = {(7, 5), (7, 4), (7, 2), (8, 5), (8, 4), (8, 2)}

H X G = {5, 4, 2} X {7, 8}

                   = {(5, 7), (5, 8), (4, 7), (4, 8), (2, 7), (2, 8)}

Q – 4 – State whether each of the following statements are true or false. If the statement is false, rewrite the given statement correctly.

(i)If P = {m, n} and Q = {n, m}, then P X Q = {(m, n), (n, m)}

Solution:) = 2 and n(Q) = 2

Therefore, n(P X Q) = 4, and here given n(PXQ) = 2,

Hence the given statement is false.

Now,

P X Q = {(m, n), (m, m), (n, n), (n, m)

(ii)If A and B are non-empty sets, the A x B is a non-empty set of ordered pairs (x, y)

such that

Solution:

False

Because if A and B are non empty sets, therefore, A X B is a non-empty set of ordered

pairs (x, y) such that

(iii) If A = {1, 2}, B = {3, 4}, then

Solution:

True

Q - 5- If A = {-1, 1}, find A x A x A.

Solution:

Given, A = {-1, 1}

Therefore,

A x A x A = {(-1, -1, -1), (-1, -1, 1), (-1, 1, 1), (1, -1, -1), (1, -1, 1), (1,-1,-1), (-1, 1, -1), (1,1,1)}

Q - 6 -

Solution:

Q – 7 – Let A = {1,2}, B = {1 ,2, 3, 4}, C = {5, 6} and D = {5, 6, 7, 8}. Verify that

Solution:

Given,

Let A = {1,2}, B = {1 ,2, 3, 4}, C = {5, 6} and D = {5, 6, 7, 8}

Therefore,

Hence,

Now,

A X B = {1, 2} x { 1, 2, 3, 4}

           = {(1, 1), (1, 2), (1, 3), (1, 4), (2, 1), (2, 2), (2, 3), (2, 4)}

And

A X C = {1, 2} x {5, 6}

           = {(1, 5), (1, 6), (2, 5), (2, 6)}

Now,

Therefore, L.H.S. = R.H.S.

Hence,

(ii) A x C is a subset of B x D

Solution:

Given, Let A = {1,2}, B = {1 ,2, 3, 4}, C = {5, 6} and D = {5, 6, 7, 8}

Therefore,

A x C = {1, 2} x {5, 6}

= {(1, 5), (1, 6), (2, 5), (2, 6)}

B x D = {1, 2, 3, 4} x {5, 6, 7, 8}

= {(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)}

As every element of Ax B is the element of B x D, so it is clear that

Ax B is a subset of B x D

Q – 8 – Let A = {1, 2} and B = {3, 4}. Write A x B. How many subsets will A x B have? List them.

Solution:

Given,

A = {1, 2} and B = {3, 4}

Therefore, A x B = {(1, 3), (1, 4), (2, 3), (2, 4)}

We know that number of subset having n element = 2ⁿ.

In A x B, n = 4

Therefore subset of A x B = 2⁴ = 16

Hence,

A x B = {(1, 3), (1, 4), (2, 3), (2, 4)}

Number of subset in AxB = 16

Q – 9 – Let A and B be two sets such that n(A) = 3 and n(B) = 2. If (x,1) (y,2), (z,1) are in A x B, find A and B where x, y, and z are distinct elements.

Solution:

Given

n(A)=3 and n(B)=2

Therefore, n(AxB) = n(A)

n(B)= 3 x 2 = 6

Now, if (x,1) (y,2), (z,1) are in A x B,

Therefore, A = {x, y, z} and B ={1, 2}

Q – 10 – The Cartesian product A x A has 9 elements among which are found (-1, 0) and (0, 1). Find the set A and the remaining elements of A x A.

Solution:

Given,

Cartesian product A x A has 9 elements among which are found (-1, 0) and (0, 1)

This means A has 3 elements as (-1,0)∈A×A and (0,1)∈A×A

Therefore, (-1,0)∈A×A⇒-1,0 ∈A

And (0,1)∈A×A⇒0,1 ∈A

Therefore, A={-1,0,1}

Hence, rest elements of A×A={(-1,-1),(-1,1),(0,-1),(0,0),(1,-1),(1,0),(1,1)