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Relations and Functions Questions and Answers
1. Find the values of x and y, if (x + 4, y – 8) = (8, 1).
Solution:
Given that (x + 4, y – 8) = (8, 1)
Compare the elements of the given ordered pairs.
Firstly, compare the first components of the given ordered pairs.
x + 4 = 8
x = 8 – 4 = 4
So, x = 4.
Now, compare the second components of the given ordered pairs.
y – 8 = 1
y = 8 + 1 = 9
So, y = 9.
Therefore, the value of x = 4 and y = 9.
2. If (x/5 + 3, y – 5/7) = (4, 5/14), find the values of x and y.
Solution:
Given that (x/5 + 3, y – 5/7) = (4, 5/14)
Compare the elements of the given ordered pairs.
Firstly, compare the first components of the given ordered pairs.
x/5 + 3 = 4
x/5 = 4-3
Therefore, x/5 = 1
x = 1 * 5 = 5
So, x = 5.
Now, compare the second components of the given ordered pairs.
y – 5/7 = 5/14
y = 5/14 + 5/7 = 15/14
So, y = 15/14.
Therefore, the value of x = 5 and y = 15/14.
3. If X = {m, n, o} and Y = {u, v}, find X × Y and Y × X. Are the two products equal?
Solution:
Given that X = {m, n, o} and Y = {u, v},
Let’s find the X × Y
X × Y = {(m, u); (m, v); (n, u); (n, v); (o, u); (o, v)}
Now, find Y × X.
Y × X = {(u, m); (u, n); (u, o); (v, m); (v, n); (v, o)}
Compare the elements of the given ordered pairs X and Y.
X × Y not equal to Y × X
Therefore, it is clearly stated that the two products are not equal.
4. If P × Q = {(x, 7); (x, 9); (y, 7); (y, 9); (z, 7); (z, 9)}, find P and Q.
Solution:
Given that P × Q = {(x, 7); (x, 9); (y, 7); (y, 9); (z, 7); (z, 9)},
We know that P is a set of all first entries in ordered pairs in P × Q.
Q is a set of all second entries in ordered pairs in P × Q.
Therefore, P = {x, y, z}
Q = {7, 9}
Therefore, the final answer is P = {x, y, z} and Q = {7, 9}
5. If M and N are two sets, and M × N consists of 6 elements: If three elements of M × N are (8, 4) (7, 3) (6, 3). Find M × N.
Solution:
Given that M and N are two sets, and M × N consists of 6 elements: If three elements of M × N are (8, 4) (7, 3) (6, 3).
We know that M is a set of all first entries in ordered pairs in M × N.
N is a set of all second entries in ordered pairs in M × N.
Therefore, M = {8, 7, 6}, and N = {4, 3}
Now, M × N = {(8, 4); (8, 3); (7, 4); (7, 3); (6, 4); (6, 3)}
Thus, M × N contains six ordered pairs.
6. If A × B = {(m, 3); (m, 7); (m, 4); (n, 3); (n, 7); (n, 4)}, find B × A.
Solution:
Given that A × B = {(m, 3); (m, 7); (m, 4); (n, 3); (n, 7); (n, 4)},
We know that A is a set of all first entries in ordered pairs in A × B.
B is a set of all second entries in ordered pairs in A × B.
Therefore, A = {m, n}, and B = {3, 7, 4}
Now, B × A = {(3, m); (3, n); (7, m); (7, n); (4, m); (4, n)}
Therefore, the final answer is B × A = {(3, m); (3, n); (7, m); (7, n); (4, m); (4, n)}
7. If P = { 2, 1, 9} and Q = {4, 5}, then
Find: (i) P × Q (ii) Q × P (iii) P × P (iv) (Q × Q)
Solution:
Given that P = { 2, 1, 9} and Q = {4, 5}
(i) P × Q = {(2, 4); (2, 5); (1, 4); (1, 5); (9, 4); (9, 5)}
(ii) Q × P = {(4, 2); (4, 1); (4, 9); (5, 2); (5, 1); (5, 9)}
(iii) P × P = {(2, 2); (2, 1); (2, 9); (1, 2); (1, 1); (1, 9); (9, 2); (9, 1); (9, 9)}
(iv) (Q × Q) = {(4, 4); (4, 5); (5, 4); (5, 5)}
8. If P = {3, 5, 7} and Q = {2, 3, 6}, state which of the following is a relation from P to Q.
(a) R₁ = {(3, 5); (6, 7); (7, 2)} (b) R₂ = {(3, 3); (7, 6)}
(c) R₃ = {(3, 2); (5, 6); (6, 7)} (d) R₄ = {(7, 2); (7, 6); (5, 3); (3, 3), (5, 2), (5, 7)}
Solution:
Given that P = {3, 5, 7} and Q = {2, 3, 6}
Note: Every element of set P is associated with a unique element of set Q. No element of P must have more than one image.
(a) f(1) = 3 and f(1) = 5 are not possible. so, this relation is not mapping from P to Q.
(b) R₂ = {(3, 3); (7, 6)}. Every element of set P is associated with a unique element of set Q. hence, it is relation from P to Q.
(c) R₃ = {(3, 2); (5, 6); (6, 7)} it’s not relation from P to Q.
(d) R₄ = {(7, 2); (7, 6); (5, 3); (3, 3), (5, 2), (5, 7)} Every element of set P is associated with a unique element of set Q. hence, it is relation from P to Q.
Therefore, the final answer is (b) R₂ = {(3, 3); (7, 6)} and (d) R₄ = {(7, 2); (7, 6); (5, 3); (3, 3), (5, 2), (5, 7)}
9. Write the domain and range of the following relations.
(a) R₁ = {(5, 4); (7, 9); (5, 9); (1, 8); (8, 6); (1, 9)}
(b) R₂ = {(p, 3); (q, 4); (r, 3); (p, 4); (s, 5); (q, 5)}
Solution:
Given that (a) R₁ = {(5, 4); (7, 9); (5, 9); (1, 8); (8, 6); (1, 9)}
(b) R₂ = {(p, 3); (q, 4); (r, 3); (p, 4); (s, 5); (q, 5)}
(a) R₁ = {(5, 4); (7, 9); (5, 9); (1, 8); (8, 6); (1, 9)}
From the given information, the Domain = {1, 5, 7, 8} and Range = {4, 6, 8, 9}
(b) R₂ = {(p, 3); (q, 4); (r, 3); (p, 4); (s, 5); (q, 5)}
From the given information, the Domain = {p, q, r, s} and Range = {3, 4, 5}
10. Let P = {3, 4, 5, 6, 7, 8}. Define a relation R from A to A by R = {(x, y) : y = x + 1}.
- Depict this relation using an arrow diagram.
- Write down the domain and range of R.
Solution:
Given that P = {3, 4, 5, 6, 7, 8}. Define a relation R from A to A by R = {(x, y) : y = x + 1}.
If x = 3, y = x + 1 = 3 + 1 = 4.
x = 4, y = x + 1 = 4 + 1 = 5.
x = 5, y = x + 1 = 5 + 1 = 6.
x = 6, y = x + 1 = 6 + 1 = 7.
x = 7, y = x + 1 = 7 + 1 = 8.
x = 8, y = x + 1 = 8 + 1 = 9.
R = {(3, 4); (4, 5); (5, 6); (6, 7); (7, 8)} where P = {3, 4, 5, 6, 7, 8}.
Domain = Set of all first elements in a relation = {3, 4, 5, 6, 7}
Range = Set of all second elements in a relation = {4, 5, 6, 7, 8}
11. Adjoining figure shows a relationship between the set P and Q. Write this relation in the roster form. What are its domain and range?
Solution:
The relation mentioned in the figure shows, P a domain and Q as a range.
Let the relation be R.
In roster form R = {(3, 6); (6, 12); (9, 18)}
Domain = Set of all first elements in a relation = {3, 6, 9}
Range = Set of all second elements in a relation = {6, 12, 18}
12. In the given ordered pairs (2, 4); (4, 16); (5, 7); (1, 3); (6, 36); (2, 9); (1, 1), find the following relationship:
(a) Is a factor of ….
(b) Is a square root of …..
Also, find the domain and range in each case.
Solution:
Given that the ordered pairs (2, 4); (4, 16); (5, 7); (1, 3); (6, 36); (2, 9); (1, 1).
(a) Is a factor of ….
Let’s find out the factor of …. from the given order pars.
(2, 4); (4, 16); (1, 3); (6, 36); (1, 1)
Domain = Set of all first elements in a relation = {1, 2, 4, 6}
Range = Set of all second elements in a relation = {1, 3, 4, 16, 36}
(b) Is a square root of …..
Let’s find out the square root of …. from the given order pars.
(2, 4); (4, 16); (6, 36).
Domain = Set of all first elements in a relation = {2, 4, 6}
Range = Set of all second elements in a relation = {4, 16, 36}
13. Draw the arrow diagrams to represent the following relations.
(a) R₁ = {(2, 2); (2, 7); (2, 8); (6, 9); (7, 4)}
(b) R₂ = {(5, 11); (5, 14); (5, 17); (6, 14); (7, 17)}
(c) R₃ = {(3, 4); (4, 6); (5, 8); (6, 10); (7, 12)}
(d) R₄ = {(a, x); (a, y); (b, p); (b, z); (c, y)}
Solution:
(a) Given that R₁ = {(2, 2); (2, 7); (2, 8); (6, 9); (7, 4)}
Let the two sets are P and Q.
The required diagram is
(b) Given that R₂ = {(5, 11); (5, 14); (5, 17); (6, 14); (7, 17)}
Let the two sets are P and Q.
The required diagram is
(c) Given that R₃ = {(3, 4); (4, 6); (5, 8); (6, 10); (7, 12)}
Let the two sets are P and Q.
The required diagram is
(d) Given that R₄ = {(a, x); (a, y); (b, p); (b, z); (c, y)}
Let the two sets are P and Q.
The required diagram is
14. Represent the following relation in the roster form.
(a) 
(b) 
(c) 
(d) 
Solution:
(a) R = {(a, x) (a, z) (b, y) (c, x) (c, q) (d, z)}
(b) R = {(3, 7) (3, 9) (4, 7) (4, 10) (5, 9) (3, 11)}
(c) R = {(2, 2) (5, 3) (10, 4) (17, 5)}
(d) R = {(11, 3) (11, 6) (13, 3) (13, 4) (13, 5) (16, 4) (16, 6) (26, 6)}