Q5. Is each relation reflexive, symmetric or transitive? (i) R = {(x, y) : x − y is an integer} in Z. In the set of people in a town at a particular time, R = {(x, y) : …}, where (a) x and y work at the same place; (b) x and y live in the same locality; (c) x is exactly 7 cm taller than y; (d) x is wife of y; (e) x is father of y (NCERT Exercise 1.1, Q1 (iv) and (v)).
(i) All three: on Z, every difference is an integer, so this is the universal relation.
For the town questions, the answers depend on the actual people and the precise meaning of each relation. Under the exercise’s intended assumptions:
(a) and (b) are equivalence relations if each person has exactly one assigned workplace and one locality. With multiple workplaces or no workplace, “share a workplace” need not have these properties.
(c) is not reflexive. It is not symmetric if a pair differing by 7 cm exists, and not transitive if a chain of two such pairs exists, since the total difference is 14 cm. If there is no such chain, it is vacuously transitive.
(d) In the textbook’s husband-and-wife model, a wife relates to her husband, who is not himself a wife: there is no two-step chain, so the relation is transitive, not reflexive, and not symmetric when at least one married pair is present. This conclusion uses that model; it is not a universal claim about all marriages.
(e) With the ordinary biological-parent interpretation, it is not reflexive, and not symmetric when a father-child pair is present. It fails transitivity when the town contains a grandfather-father-child chain. Without a two-step chain it is vacuously transitive.
Q6. Show that R = {(a, b) : a ≤ b} in R is reflexive and transitive but not symmetric. Then check R = {(a, b) : a ≤ b3} in R (NCERT Exercise 1.1, Q4 and Q5).
For a ≤ b: a ≤ a; 1 ≤ 2 but 2 ≤ 1 is false; a ≤ b and b ≤ c give a ≤ c. For a ≤ b3: not reflexive, since 1/2 > (1/2)3 = 1/8; not symmetric, since 1 ≤ 23 but 2 > 13; not transitive, since 3 ≤ (3/2)3 = 3.375 and 3/2 ≤ (6/5)3 = 1.728, but 3 > 1.728. It has none of the three.
Q7. Show that R = {(a, b) : |a − b| is even} in A = {1, 2, 3, 4, 5} is an equivalence relation, that the elements of {1, 3, 5} are all related to each other, as are those of {2, 4}, and that no element of {1, 3, 5} is related to any element of {2, 4} (NCERT Exercise 1.1, Q8).
|a − a| = 0 is even; |b − a| = |a − b|; and if a − b and b − c are even, so is their sum a − c. A difference is even exactly when the two numbers are both odd or both even, so the classes are {1, 3, 5} and {2, 4}, and an odd and an even number are never related.
Q8. Give an example of a relation in {1, 2, 3} that is (i) symmetric but neither reflexive nor transitive; (ii) transitive but neither reflexive nor symmetric; (iii) reflexive and symmetric but not transitive; (iv) reflexive and transitive but not symmetric; (v) symmetric and transitive but not reflexive (NCERT Exercise 1.1, Q10).
One answer each (others are possible): (i) {(1, 2), (2, 1)}; (ii) {(1, 2)}; (iii) {(1, 1), (2, 2), (3, 3), (1, 2), (2, 1), (2, 3), (3, 2)}, where (1, 2), (2, 3) are in but (1, 3) is not; (iv) {(1, 1), (2, 2), (3, 3), (1, 2)}; (v) {(1, 1), (1, 2), (2, 1), (2, 2)}, which leaves out (3, 3).
Q9. In the set of points of a plane, P R Q when P and Q are at the same distance from the origin. Show that R is an equivalence relation, and that the points related to a point P ≠ (0, 0) form the circle through P with centre at the origin (NCERT Exercise 1.1, Q11).
“Has the same distance from O as” is reflexive, symmetric and transitive, because equality of numbers is. If OP = r > 0, the points related to P are exactly the points at distance r from O: the circle with centre O and radius r, which passes through P.
Q10. Show that “x and y have the same number of pages” on the books of a library, and “P and P have the same number of sides” on the set of all polygons, are equivalence relations. Which polygons are related to the right triangle with sides 3, 4 and 5? (NCERT Exercise 1.1, Q7 and Q13)
Each relation says two objects have the same value of something (pages, sides), and equality is reflexive, symmetric and transitive. The polygons related to the 3, 4, 5 triangle are all the polygons with 3 sides: the set of all triangles.
Q11. R = {(a, b) : a = b − 2, b > 6} in N. Which is true? (A) (2, 4) ∈ R; (B) (3, 8) ∈ R; (C) (6, 8) ∈ R; (D) (8, 7) ∈ R (NCERT Exercise 1.1, Q16).
(2, 4) fails b > 6; 3 ≠ 8 − 2; 6 = 8 − 2 and 8 > 6; 8 ≠ 7 − 2. The answer is (C).
Q12. (a) If R and R are equivalence relations in A, show that R ∩ R is one too (NCERT Miscellaneous Example 18, page 13). (b) For a non-empty set X, is “A ⊂ B” an equivalence relation on the power set P(X)? (NCERT Miscellaneous Exercise, Q3)
(a) Each (a, a) is in both, so in R ∩ R. If (a, b) is in both, so is (b, a). If (a, b) and (b, c) are in both, so is (a, c). The union need not be one: in {1, 2, 3}, the equivalence relations with classes {1, 2}, {3} and {1}, {2, 3} have a union containing (1, 2) and (2, 3) but not (1, 3). (b) No. It is reflexive (A ⊂ A, with ⊂ allowing equality as in NCERT) and transitive, but not symmetric: φ ⊂ X while X ⊄ φ, since X is non-empty.
Q13. Let A = {1, 2, 3}. How many relations in A contain (1, 2) and (1, 3) and are reflexive and symmetric but not transitive? (A) 1 (B) 2 (C) 3 (D) 4 (NCERT Miscellaneous Exercise, Q6)
Reflexive and symmetric forces {(1, 1), (2, 2), (3, 3), (1, 2), (2, 1), (1, 3), (3, 1)}. This is not transitive: (2, 1), (1, 3) needs (2, 3). The only pairs left are (2, 3) and (3, 2), and symmetry adds them together, which gives A × A, a transitive relation. So exactly one relation works: (A).
Q14. Let A = {1, 2, 3}. Show that exactly three relations in A contain (1, 2) and (2, 3) and are reflexive and transitive but not symmetric (NCERT Miscellaneous Example 23, page 14).
Reflexive and transitive forces R = {(1, 1), (2, 2), (3, 3), (1, 2), (2, 3), (1, 3)}. The pairs left are (2, 1), (3, 2) and (3, 1). Adding (2, 1) alone, or (3, 2) alone, keeps R transitive and not symmetric. Adding (3, 1), or both (2, 1) and (3, 2), forces every remaining pair by transitivity, giving the symmetric A × A. So the relations are R, R ∪ {(2, 1)} and R ∪ {(3, 2)}: three.
Q15. A set has 3 elements. How many relations in it are (i) symmetric; (ii) both reflexive and symmetric? (iii) How many equivalence relations does a set of 4 elements have?
(i) 3 diagonal pairs and 3 mirror couples, 6 free choices: 26 = 64. (ii) The diagonal is fixed, so only the 3 couples are free: 23 = 8. (iii) One for each way of splitting the 4 elements into classes: one class (1 way), 3 + 1 (4 ways), 2 + 2 (3 ways), 2 + 1 + 1 (6 ways) and four single classes (1 way), 15 in all.