AReflexive and Symmetric
BSymmetric and Transitive
CTransitive and Reflexive
DAll of the above
Answer:
C. Transitive and Reflexive
Read Explanation:
Understanding the "Division" Relation
The relation "division" on the set of positive integers (denoted as ℤ+ or {1, 2, 3, ...}) means that for any two positive integers 'a' and 'b', 'a divides b' (written as a | b) if and only if there exists a positive integer 'k' such that b = a * k.
Properties of Relations in Set Theory
To analyze the "division" relation, we examine its fundamental properties:
Reflexivity: A relation R on a set A is reflexive if for every element a ∈ A, (a, a) ∈ R.
Symmetry: A relation R on a set A is symmetric if whenever (a, b) ∈ R, then (b, a) ∈ R.
Anti-symmetry: A relation R on a set A is anti-symmetric if whenever (a, b) ∈ R and (b, a) ∈ R, then a = b.
Transitivity: A relation R on a set A is transitive if whenever (a, b) ∈ R and (b, c) ∈ R, then (a, c) ∈ R.
Why "Division" is Reflexive
For any positive integer 'a', 'a divides a' is always true. This is because 'a' can be written as 'a * 1', and 1 is a positive integer.
Example: 7 divides 7, as 7 = 7 * 1.
Therefore, the "division" relation on the set of positive integers is Reflexive.
Why "Division" is Transitive
Consider three positive integers a, b, and c.
If 'a divides b' (a | b), it means b = a * k₁ for some positive integer k₁.
If 'b divides c' (b | c), it means c = b * k₂ for some positive integer k₂.
Substituting the first equation into the second: c = (a * k₁) * k₂ = a * (k₁ * k₂).
Since k₁ and k₂ are positive integers, their product (k₁ * k₂) is also a positive integer.
This demonstrates that 'a divides c' (a | c).
Example: If 2 divides 6 (6 = 2*3) and 6 divides 18 (18 = 6*3), then 2 divides 18 (18 = 2*9).
Therefore, the "division" relation is Transitive.
Why "Division" is NOT Symmetric
If 'a divides b', it does not necessarily mean 'b divides a'.
Example: 2 divides 4 (since 4 = 2*2). However, 4 does not divide 2, as there is no positive integer 'k' such that 2 = 4 * k.
Thus, the "division" relation is Not Symmetric.
Why "Division" IS Anti-symmetric
If 'a divides b' (b = a * k₁) and 'b divides a' (a = b * k₂) for positive integers a, b, k₁, k₂.
Substituting 'a' into the first equation: b = (b * k₂) * k₁ ⇒ b = b * (k₁ * k₂).
Since b is a positive integer, we can divide by b, which gives 1 = k₁ * k₂.
As k₁ and k₂ are positive integers, the only possibility for their product to be 1 is if k₁ = 1 and k₂ = 1.
If k₁ = 1, then b = a * 1, implying b = a.
Therefore, the "division" relation on positive integers is Anti-symmetric.