21.4. Question 3

Now in question 3, there are several parts and, you are given a set S, consisting of n elements where n is not 0. So, implicitly I am assuming here that n is greater than equal to 1. And now I have to count various number possible relations satisfying some properties.

In this section, the task is to analyze how many different kinds of relations can be formed with a set S that consists of n elements, where n is a positive integer. Relations can have specific properties or characteristics, and we will investigate these.

Think of a set S as a group of friends, where each friend can have relationships with one another. We want to categorize these relationships based on certain rules, like friendship (symmetric), where if one friend considers another a friend, the feeling is mutual.

The first property here is I am interested to count the number of relations on this set S which are symmetric. Just to recall that definition of a symmetric relation is that if you have the element or the ordered pair (i, j) present in the relation R then the ordered pair (j, i) should also be present in the relation R.

A symmetric relation is one where if one element is related to another, then the reverse must also hold true. For example, if Alice is friends with Bob, then Bob must also be friends with Alice. To count how many symmetric relations we can form, we look at all possible pairs of elements from set S and consider each pair's reverse.

Imagine a friendship circle where if Person A invites Person B, then Person B also invites Person A. The symmetric relation ensures that both individuals have acknowledged their friendship by inviting one another.

So, what I have done here is I have drawn all possible n^2 ordered pairs, these are the ordered pairs in the set S x S. So, you can have n^2 such ordered pairs and any subset of these n^2 ordered pairs will constitute a relation over the set S.

We can form a relation using subsets of the ordered pairs from the Cartesian product of S with itself (S x S). Since there are n elements in S, there are n² possible pairs. We need to select subsets that respect the property of symmetry, meaning if we pick (i, j), we must also include (j, i) for the relation to be symmetric.

This is like creating a list of pairs for a school project where students can work with others. If Jamie works with Alex, Alex should also be listed as working with Jamie to keep the teamwork balanced.

So, my claim is that if you take the union of the ordered pairs in the subset A and the ordered pairs in A^{-1} then the collection of these ordered pairs will give you a symmetric relation.

To ensure a relation is symmetric, for every pair (i, j) we include in our set, we also include its reverse (j, i). We can choose subsets of one direction and guarantee the opposite forms part of the relation by including the inverse pairs.

Think of relationships in a community where if one person mentions they will help another, the second should also have the understanding that the first will be available for help—a mutual agreement represented in both directions.

Now in part b of the question I want to find out the number of relations over the set S, which are anti-symmetric. Just to recap, this is the requirement from an anti-symmetric relation...

An anti-symmetric relation allows for (a, b) and (b, a) to exist only if a equals b. This means distinct pairs cannot form two-way relationships simultaneously. For example, if (1, 2) is in the relation, (2, 1) cannot be.

Imagine a ranking system in a competition. If contestant A beats contestant B, then contestant B cannot also beat contestant A; only one can rank higher unless they are tied, enforcing this anti-symmetric nature of relationships.

So, let us focus on the non-diagonal ordered pairs of the form (i, j) and (j, i) where i and j are distinct...

We analyze how many ways to form non-diagonal pairs of relations while adhering to anti-symmetric conditions. For each distinct (i, j), we can either include none, only (i, j), or (j, i) but not both. This creates a decision-making space in our relation creation.

Visualize making decisions with friends about who will pick the restaurant to eat at. If one friend suggests a place, the other cannot suggest the same place back unless they're in agreement (a tie), fostering independent decision-making.

In part C of the third question, you are supposed to find out the number of relations which are asymmetric...

An asymmetric relation prohibits (a, b) from appearing if (b, a) is also present. None of the diagonal pairs can be in an asymmetric relation. Thus, we explore how relationships can exist independently without reciprocal acknowledgment.

Think of a teacher-student relationship where a teacher may assign tasks but the student does not have any obligation to reciprocate the relationship or actions, reflecting the one-directionality of an asymmetric relation.

In part d, I am interested to find out how many relations I can form which are irreflexive...

An irreflexive relation means none of the pairs of the form (a, a) can be included in the relation. However, pairs where elements are different can be included freely, giving flexibility in selecting relations while maintaining this irreflexivity.

Imagine a sports team rule where players cannot score against themselves — this ensures that all interactions are with others, not with their own actions.

Part e, I now want to find the number of relations which are simultaneously reflexive as well as symmetric...

For a relation to be both reflexive and symmetric, we must include all diagonal pairs and decide on the pairs that maintain symmetry without disallowing the existence of both (a, b) and (b, a). The inclusion of all diagonal pairs establishes reflexivity, whilst the relationships must adhere to symmetrical conditions based on choice.

Consider a situation in a social circle where a friend always acknowledges the birthdays of all friends — they must send invitations (reflexive) and respond to invitations received (symmetric).

The last part of the question, I have to find out the number of relations, which are neither reflexive nor irreflexive...

This question requires us to define conditions whereby at least one pair is not reflexive, while at least one is reflexive. By analyzing the set conditions of reflexivity and irreflexivity and considering total possibilities reduced by existing conditions, the relations can be calculated logically.

Imagine a classroom where some students do not raise their hands while participating. These actions demonstrate a lack of reflexivity for some (not all participate), while others might consistently participate (showing reflexivity).

--