91Ó°ÊÓ

Reflexivity is like looking into a mirror and seeing oneself.
For a relation to be reflexive, every single element must be related to itself. Think of it as the condition where each individual in a group can say they are equal to themselves.
In our exercise, we explore this with integers from the set \(\mathbb{Z}\), where the relation is defined such that \(a \sim b\) if and only if \(a-b\) is even.
To check reflexivity, let's see if each element \(a\) relates to itself; this means examining \(a \sim a\).
Mathematically, we have \(a-a = 0\), and since 0 is an even number, this condition is satisfied. Thus, each element is indeed related to itself, fulfilling reflexivity.
This property is crucial because it signifies that every element is in agreement with itself under the given relation.

The symmetry property in a relation is akin to a two-way street.
If a relation is symmetric, it means the relationship works both ways. If element \(a\) is related to \(b\), then \(b\) must also be related to \(a\).
In mathematical terms, for our set \(\mathbb{Z}\), let's take \(a \sim b\). This means \(a-b\) is an even number, implying \(a-b = 2k\) for some integer \(k\).
Symmetry demands that \(b \sim a\) too.
Let's see if this holds true: if we take \(b-a\), it equals \(-2k\), which remains an even number.
Thus, symmetry is maintained because the even nature of the difference doesn't change, regardless of the direction of subtraction.
This property shows the mutual relationship between pairs in the set according to the defined relation.

Think of transitivity as a relational domino effect.
It ensures that if a relation links two pairs, it also links the first and the last in the sequence. In a simple sense, if \(a\) is related to \(b\), and \(b\) is related to \(c\), then \(a\) must relate to \(c\) to satisfy transitivity.
Here, in our case on \(\mathbb{Z}\), suppose \(a \sim b\) and \(b \sim c\). This means \(a-b = 2k\) and \(b-c = 2m\) for some integers \(k\) and \(m\).
For transitivity to hold, \(a \sim c\) must also be true. By adding the equations, \(a-c = (a-b) + (b-c) = 2k + 2m = 2(k+m)\), which is clearly even.
Therefore, transitivity is confirmed as the pathway from \(a\) to \(c\) through \(b\) remains even.
This property ensures cohesion and linkage within the set under the given relation.