Question

Consider the following rules on the set of natural numbers n and say if the resulting relation is transitive or not. and why. (a) "is less than or equal to" (b) "is a factor of" (c) "is 3 less than"

Options:
A) (a) Transitive, (b) Not transitive, (c) Transitive
B) (a) Transitive, (b) Transitive, (c) Not transitive
C) (a) Not transitive, (b) Transitive, (c) Transitive
D) (a) Not transitive, (b) Not transitive, (c) Not transitive

Answer 1

The relations "is less than or equal to" and "is a factor of" are transitive because they satisfy the definition of a transitive relation. However, the relation "is 3 less than" is not transitive because it does not fulfill the requisite conditions for transitivity.

Step-by-step explanation:

The question asks whether three different relations in the set of natural numbers are transitive or not. A relation is considered transitive if, whenever one element is related to a second, and the second is related to a third, the first is also related to the third.

1. "is less than or equal to" is a classic example of a transitive relation. If a ≤ b and b ≤ c, then it must be true that a ≤ c.
2. "is a factor of" is also a transitive relation because if a is a factor of b, and b is a factor of c, then a must be a factor of c as well.
3. However, "is 3 less than" is not transitive. For example, if a is 3 less than b (a = b - 3) and b is 3 less than c (b = c - 3), a will actually be 6 less than c (a = c - 6), not 3.

Therefore, the correct answer is Option B: (a) Transitive, (b) Transitive, (c) Not transitive.