Part C: Asymmetric Relations

Today, we are going to explore asymmetric relations. An asymmetric relation means that if (a, b) is in the relation, then (b, a) cannot be. This is different from symmetric relations where both can exist.

Can you give an example of an asymmetric relation?

Of course! Consider the relation 'is less than' on the set of real numbers. If a < b holds, then b < a cannot hold.

Are all relations symmetric or asymmetric?

Great question! Some relations can be neither, while others can be both under certain conditions. We are going to analyze those conditions next.

To help remember this, think of the acronym 'ASYMM', which stands for 'A Set Yields Many Misinterpretations' in terms of conditions applied to sets.

In summary, asymmetric relations have a strict one-directional property that cannot coexist with its reverse.

Let’s move on to counting relations. When we have a set S of n elements, how many asymmetric relations can we create?

Do we consider the pairs (i, j) and (j, i) together in this counting?

Exactly! When you include one, the other cannot be allowed in an asymmetric relation. So, we will only look at distinct pairs.

And what about the diagonal pairs, like (i, i)?

Good point! Diagonal pairs must be excluded in an asymmetric relation. We have a total of n² pairs, minus n diagonal pairs to form non-diagonal pairs.

Remember, for each pair (i, j), there are three choices: include (i, j), include neither, or include (j, i) only without violating the asymmetric rule.

Thus, the total number of asymmetric relations can be expressed mathematically!

Now that we have laid the foundation, let’s practice with some examples. If we have 4 elements in set S, how many asymmetric relations could we establish?

We have 4 × 3 non-diagonal pairs, right?

Correct! Which gives you 12 non-diagonal pairs to work with.

So, it's 3 options per pair, does that mean we take 3 to the power of the number of non-diagonal pairs?

Yes! That’s the right formula; thus it results in a potential calculation of 3^(12) for your specific example. Very good!

In summary: we can define relations mathematically but always need to pay attention to conditions that apply.

Now we will focus on mathematical proofs for these relations. Let's consider the implications of 'if x is in A and implies x in B', what can we conclude?

That would mean A ⊆ B, right? But how do we prove that?

Great observation! We can use a direct proof by taking an arbitrary element. If x is in A, and we satisfy (A ∩ C) ⊆ (B ∩ C), it holds that x must also be in B.

What if we change C? Does it still hold?

Yes, the universally quantified property should hold for any set C chosen, that’s what makes it robust.

To summarize: proofs establish the foundation for understanding relations on a more logical basis.