[CBSE 2014)
11. Is it true that every relation which is symmetric and transitive is also reflexive? Give
reasons.
12. An integer m is said to be related to another integer n if m is a multiple of n. Check if the
relation is symmetric, reflexive and transitive.
13. Show that the relation">" on the set R of all real numbers is reflexive and transitive but not
symmetric.
14. Give an example of a relation which is
(i) reflexive and symmetric but not transitive.
(ii) reflexive and transitive but not symmetric.
(iii) symmetric and transitive but not reflexive.
(iv) symmetric but neither reflexive nor transitive.
(v) transitive but neither reflexive nor symmetric.
INCERT)
Answers
Answer:
11. No, it is not true. The relation R on A is symmetric and transitive. However, it is not reflexive. Hence, R is not reflexive.
12. en this relation is reflexive as every integer is a multiple of its own i.e. (a,a)∈R, ∀a∈Z ( set of integers).
Again this relation is not symmetric since 4 is a multiple of 2 but 2 is not a multiple of 4 i.e. (4,2)∈R but (2,4)
∈R.
Also this relation is transitive as if a is a multiple of b and b is a multiple of c then naturally a is a multiple of c i.e. (a,b)∈R,(b,c)∈R⇒(a
13. Reflexivity : Let a be an arbitrary element of R. Hence, R is reflexive. Hence, R is not symmetric .
14. Relation R is symmetric since (a, b) ∈ R ⇒ (b, a) ∈ R for all a, b ∈ R. Relation R is not transitive since (4, 6), (6, 8) ∈ R, but (4, 8) ∉ R. Hence, relation R is reflexive and symmetric but not transitive. Clearly (a, a) ∈ R as a3 = a3.
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