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Learn it your way
A short class with Sarah and her students. The other ways are under More.
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Welcome, everyone! Today, we're going to start by discussing how to prove that if (A ∩ C) is a subset of (B ∩ C) for any set C, then A must be a subset of B. Can anyone tell me what a subset is?
A subset is a set where all its elements are also contained in another set.
Exactly! Now, if we substitute C with A, what does that give us?
It gives us A ∩ A, which is just A, making it A ⊆ B ∩ A.
Great! Therefore, any element x in A suggests that it must be in B as well. With that in mind, let's continue building on similar principles. Remember, the acronym PIM for 'Proof Involves Manipulation' to help remember this approach.
Let's explore the implications surrounding sets A, B, and C further. If x is in A and we have the implication that x in B leads to x in C, what do you think happens?
So, if x is in A and follows that rule, it should also be in C when it's in B.
Exactly! And thus, we prove that A intersect B is a subset of C. Always remember this: Implications build assertions of relationships between sets!
Is there a formula for counting subsets based on these properties?
Good question! We'll touch on counting later in today's tutorial, so keep that in mind.
Now, will someone define a symmetric relation for me?
If (i, j) is in a relation, then (j, i) must also be included for it to be symmetric.
Absolutely! So based on that definition, how do we form a symmetric relation from n elements?
We need to focus on pairs of ordered elements and include their inverses. What happens if we pick one from the upper triangle of a relation matrix?
Great observation! The total pairs can lead to many combinations as they are mutually inclusive in symmetric relations.
Who can explain the anti-symmetric property to me?
If both (a, b) and (b, a) are present in a relation, then a must equal b.
Exactly! So, applying this definition in counting how many relations we can form becomes crucial. Can anyone summarize how we calculate these pairs?
We consider non-diagonal pairs and ensure either or condition while excluding both in the count.
Well said! Remember, anti-symmetric relations challenge you to think critically about what pairs can coexist.
As we wrap up, can anyone recap what we learned about set relations today?
We established proofs for subset relations and the implications for whether A is a subset of B.
And we classified relations into symmetric, anti-symmetric, reflexive, and irreflexive categories.
Exactly! Also, remember how you can calculate the number of possible relations depending on the properties defined. Great job today, everyone!
To see a subset, just check the set, all elements meet, no need for a fret.
Imagine A and B are friends in a group, if C sees them both, they'll hang in a loop.
Remember 'SARA' for Symmetric, Anti-symmetric, Reflexive, and Irreflexive relations.
For relations remember 'SARA' - Symmetric, Anti-symmetric, Reflexive, and Irreflexive.
Term
What is a symmetric relation?
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Term
What does irreflexive mean?
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A set A is a subset of set B if every element of A is also an element of B.
A relation R is symmetric if for all a, b, if (a, b) is in R then (b, a) is also in R.
A relation R is anti-symmetric if for all a, b, if (a, b) and (b, a) are both in R, then a must equal b.
A relation R is irreflexive if for every element a in the set, (a, a) is not in R.
A relation R is reflexive if for every element a in the set, (a, a) is in R.
This tutorial covers various aspects of set theory and relations, including proofs of subset relations and characterization of relations such as symmetric, anti-symmetric, and irreflexive relations.
In this tutorial, we explore foundational concepts related to sets and their relations. Through various problems, we analyze and demonstrate the properties of subsets, implications of relations, and specific classifications of relations such as symmetric, anti-symmetric, and reflexive. Each problem emphasizes logical reasoning and proofs to derive key conclusions in set theory.
In Tutorial 3, we dive deeper into the properties of sets and relations using various logical and mathematical proofs. The tutorial is structured around a series of questions focused on the assertion of set relations and their implications.
Subset Relations: The first part of the tutorial asserts that if for any set C, the intersection of sets A and C is a subset of the intersection of B and C, then it can be concluded that A is a subset of B. The proof relies on substituting specific sets into the initial premise and showing that any element in A must also exist in B.
Implications of Relations: The second question investigates the relationship between sets A, B, and C, demonstrating that if an element x in A satisfies certain implications regarding B and C, then it leads to a conclusion that establishes the relation between A, B, and C.
Types of Relations: The tutorial progresses to defining various types of relations—including symmetric, anti-symmetric, irreflexive, and reflexive. For each type, the tutorial provides proofs and counts the number of possible relations based on the properties dictated by the definitions. Through these exercises, students learn how to construct and analyze relations within a set of n elements, seeking to determine how many valid subsets follow particular rules.
Counting Techniques: The problems propose calculating the total number of valid relations by breaking down possibilities for diagonal versus non-diagonal pairs and applying logical rules such as symmetry and anti-symmetry to derive the final counts.
Overall, the tutorial reinforces set theory foundations through an exploration of various logical constructs and provides procedural strategies for solving complex relational problems.