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An integer m is said to be related to another integer n if m is a multiple of n then relation is
- a)reflexive and symmetric
- b)reflexive and transitive
- c)symmetric and transitive
- d)equivalence relation
Correct answer is option 'B'. Can you explain this answer?
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An integer m is said to be related to another integer n if m is a mult...
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An integer m is said to be related to another integer n if m is a mult...
Reflexive Relation:
A relation is said to be reflexive if every element in the set is related to itself. In this case, for the relation between integers m and n, m should be a multiple of itself, which is always true. Therefore, the relation is reflexive.
Transitive Relation:
A relation is said to be transitive if for any three elements a, b, and c in the set, if a is related to b and b is related to c, then a is related to c. In this case, if m is a multiple of n, and n is a multiple of p, then m is also a multiple of p. This is true because if m is a multiple of n, it can be expressed as m = kn for some integer k. Similarly, if n is a multiple of p, it can be expressed as n = lp for some integer l. Substituting the value of n in the expression for m, we get m = k(lp) = (kl)p, which shows that m is a multiple of p. Therefore, the relation is transitive.
Symmetric Relation:
A relation is said to be symmetric if for any two elements a and b in the set, if a is related to b, then b is also related to a. However, in this case, if m is a multiple of n, it does not necessarily mean that n is a multiple of m. For example, if m = 6 and n = 2, then m is a multiple of n (6 = 3*2), but n is not a multiple of m. Therefore, the relation is not symmetric.
Equivalence Relation:
An equivalence relation is a relation that is reflexive, symmetric, and transitive. Since the given relation is not symmetric, it cannot be an equivalence relation.
Therefore, the correct answer is option B) reflexive and transitive.
A relation is said to be reflexive if every element in the set is related to itself. In this case, for the relation between integers m and n, m should be a multiple of itself, which is always true. Therefore, the relation is reflexive.
Transitive Relation:
A relation is said to be transitive if for any three elements a, b, and c in the set, if a is related to b and b is related to c, then a is related to c. In this case, if m is a multiple of n, and n is a multiple of p, then m is also a multiple of p. This is true because if m is a multiple of n, it can be expressed as m = kn for some integer k. Similarly, if n is a multiple of p, it can be expressed as n = lp for some integer l. Substituting the value of n in the expression for m, we get m = k(lp) = (kl)p, which shows that m is a multiple of p. Therefore, the relation is transitive.
Symmetric Relation:
A relation is said to be symmetric if for any two elements a and b in the set, if a is related to b, then b is also related to a. However, in this case, if m is a multiple of n, it does not necessarily mean that n is a multiple of m. For example, if m = 6 and n = 2, then m is a multiple of n (6 = 3*2), but n is not a multiple of m. Therefore, the relation is not symmetric.
Equivalence Relation:
An equivalence relation is a relation that is reflexive, symmetric, and transitive. Since the given relation is not symmetric, it cannot be an equivalence relation.
Therefore, the correct answer is option B) reflexive and transitive.
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