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"n/m" means that n is a factor of m, then the relation T is
- a)relexive and symmetric
- b)transitive and symmetric
- c)relexive, transitive and symmetric
- d)relexive, transitive and not symmetric
Correct answer is option 'D'. Can you explain this answer?
Verified Answer
"n/m" means that n is a factor of m, then the relation T isa...
′/′ is reflexive since every natural number is a factor of itself that in n/n for n∈N.
′/′ is transitive if n is a factor of m and m is a factor of P, then n is surely a factor of P.
However, ′/′ is not symmetric.
example, 2 is a factor of 4 but 4 is not a factor of 2.
′/′ is transitive if n is a factor of m and m is a factor of P, then n is surely a factor of P.
However, ′/′ is not symmetric.
example, 2 is a factor of 4 but 4 is not a factor of 2.
Most Upvoted Answer
"n/m" means that n is a factor of m, then the relation T isa...
′/′ is reflexive since every natural number is a factor of itself that in n/n for n∈N.
′/′ is transitive if n is a factor of m and m is a factor of P, then n is surely a factor of P.
However, ′/′ is not symmetric.
example, 2 is a factor of 4 but 4 is not a factor of 2.
′/′ is transitive if n is a factor of m and m is a factor of P, then n is surely a factor of P.
However, ′/′ is not symmetric.
example, 2 is a factor of 4 but 4 is not a factor of 2.
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"n/m" means that n is a factor of m, then the relation T isa...
Understanding the Relation T: n/m
The relation T defined by "n/m" means that n is a factor of m. To analyze the properties of this relation, we need to evaluate its reflexivity, symmetry, and transitivity.
Reflexivity
- A relation is reflexive if every element is related to itself.
- In this case, for any integer n, n/n holds true because n is always a factor of itself.
- Therefore, T is reflexive.
Symmetry
- A relation is symmetric if whenever n/m holds true, then m/n must also hold true.
- For example, if 2 is a factor of 4 (2/4), it does not imply that 4 is a factor of 2 (4/2).
- Thus, T is not symmetric.
Transitivity
- A relation is transitive if whenever n/m and m/p hold true, then n/p must also hold true.
- For instance, if 2 is a factor of 4 (2/4) and 4 is a factor of 8 (4/8), then 2 is indeed a factor of 8 (2/8).
- Hence, T is transitive.
Conclusion
- In summary, the relation T is:
- Reflexive: Every number is a factor of itself.
- Not symmetric: A factor relationship does not imply the reverse.
- Transitive: The factor relationship holds through chains of factors.
Therefore, the correct answer is option 'D': T is reflexive, transitive, and not symmetric.
The relation T defined by "n/m" means that n is a factor of m. To analyze the properties of this relation, we need to evaluate its reflexivity, symmetry, and transitivity.
Reflexivity
- A relation is reflexive if every element is related to itself.
- In this case, for any integer n, n/n holds true because n is always a factor of itself.
- Therefore, T is reflexive.
Symmetry
- A relation is symmetric if whenever n/m holds true, then m/n must also hold true.
- For example, if 2 is a factor of 4 (2/4), it does not imply that 4 is a factor of 2 (4/2).
- Thus, T is not symmetric.
Transitivity
- A relation is transitive if whenever n/m and m/p hold true, then n/p must also hold true.
- For instance, if 2 is a factor of 4 (2/4) and 4 is a factor of 8 (4/8), then 2 is indeed a factor of 8 (2/8).
- Hence, T is transitive.
Conclusion
- In summary, the relation T is:
- Reflexive: Every number is a factor of itself.
- Not symmetric: A factor relationship does not imply the reverse.
- Transitive: The factor relationship holds through chains of factors.
Therefore, the correct answer is option 'D': T is reflexive, transitive, and not symmetric.
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