21.1. Discrete Mathematics

Let's start with the condition (A ∩ C) ⊆ (B ∩ C). Based on this, how would we prove that A is a subset of B?

Could we begin by substituting a specific set for C?

Exactly! If we set C = A, we find that A ∩ A equals A, implying A ⊆ (B ∩ A).

So, if an element x is in A, it must also be in B since it’s in A ∩ B.

Right! Therefore, we conclude A ⊆ B. Remember, this logical flow is crucial for understanding further relations.

Can we apply this method to other set relations as well?

Yes! This type of reasoning is foundational in proving various properties in set theory.

What memory aid can we use to remember this?

An easy mnemonic is 'A is part of B (A ⊆ B)’ to remind you of the proof we went through today.

To summarize, we showed how to prove A ⊆ B using a substitution into the set intersection condition, allowing us to reason through more complex relationships later.

Now, let's explore symmetric relations. What defines a symmetric relation?

If (i, j) is in R, then (j, i) must also be in R.

Correct! And how about anti-symmetric relations?

In an anti-symmetric relation, if both (a, b) and (b, a) are present, then a must equal b.

Precisely! Let’s practice determining the conditions for forming symmetric and anti-symmetric relations.

How do we find the number of symmetric relations on a set of n elements?

Good question! We can form relations from the upper triangular part of an n x n matrix. The count of subsets from these highlights the combinations of symmetric relations.

So, should we focus on combinations to solve these?

Exactly! Let’s summarize: symmetric relations require mutual element inclusion, while anti-symmetric only allow pairs under specific conditions.

Let’s dive into counting various types of relations. What relation are we interested in when both properties are true?

That would be when we consider symmetric and anti-symmetric properties.

Correct! With both requirements, what can we conclude about the relations possible?

We can only include the diagonal elements, right?

Yes! Thus, there’s only one relation possible when both properties hold: the diagonal itself.

What if we consider irreflexive as well?

Great thought! An irreflexive relation would not allow diagonal elements, leading to zero valid relations.

Can we summarize the numbers related to these properties?

Absolutely! For relations that are both symmetric and anti-symmetric, we found just one. Conversely, for irreflexivity combined with the others, it fundamentally limits our outcomes.