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The binary relation S = f (empty set) on set A = {1,2, 3} is
- a)Neither reflexive nor symmetric
- b)Symmetric and reflexive
- c)Transitive and reflexive
- d)Transitive and symmetric
Correct answer is option 'D'. Can you explain this answer?
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Verified Answer
The binary relation S = f (empty set) on set A = {1,2, 3} isa)Neither ...
The empty relation on any set is always transitive and symmetric but not reflexive.
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The binary relation S = f (empty set) on set A = {1,2, 3} isa)Neither ...
Explanation:
Transitive Relation:
- A relation S on set A is transitive if for all a, b, c ∈ A, if (a, b) ∈ S and (b, c) ∈ S, then (a, c) ∈ S.
- In this case, since the empty set does not contain any ordered pairs, the relation S = f (empty set) is vacuously transitive.
Symmetric Relation:
- A relation S on set A is symmetric if for all a, b ∈ A, if (a, b) ∈ S, then (b, a) ∈ S.
- Since the empty set does not contain any ordered pairs, the relation S = f (empty set) is vacuously symmetric.
Reflexive Relation:
- A relation S on set A is reflexive if for all a ∈ A, (a, a) ∈ S.
- Since the empty set does not contain any elements, the relation S = f (empty set) is vacuously reflexive.
Therefore, the binary relation S = f (empty set) on set A = {1, 2, 3} is both transitive and symmetric, but not reflexive.
Transitive Relation:
- A relation S on set A is transitive if for all a, b, c ∈ A, if (a, b) ∈ S and (b, c) ∈ S, then (a, c) ∈ S.
- In this case, since the empty set does not contain any ordered pairs, the relation S = f (empty set) is vacuously transitive.
Symmetric Relation:
- A relation S on set A is symmetric if for all a, b ∈ A, if (a, b) ∈ S, then (b, a) ∈ S.
- Since the empty set does not contain any ordered pairs, the relation S = f (empty set) is vacuously symmetric.
Reflexive Relation:
- A relation S on set A is reflexive if for all a ∈ A, (a, a) ∈ S.
- Since the empty set does not contain any elements, the relation S = f (empty set) is vacuously reflexive.
Therefore, the binary relation S = f (empty set) on set A = {1, 2, 3} is both transitive and symmetric, but not reflexive.
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The binary relation S = f (empty set) on set A = {1,2, 3} isa)Neither reflexive nor symmetricb)Symmetric and reflexivec)Transitive and reflexived)Transitive and symmetricCorrect answer is option 'D'. Can you explain this answer?
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