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21.4.3. Part C: Asymmetric Relations
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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.
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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!
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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.
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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.
Overview
Short Summary
This section explores the properties of asymmetric relations, including the conditions for reflexivity, symmetry, anti-symmetry, and their implications in set theory.
Medium Summary
In this section, we delve into the definitions and properties of asymmetric relations through logical implications and examples. The focus is on how these relations can be confirmed or disproved based on algebraic expressions and set properties, such as counting symmetric, anti-symmetric, and reflexive relations.
Detailed Summary
Part C: Asymmetric Relations
In this section, we discuss the concept of asymmetric relations within set theory. Given arbitrary sets A and B, specific logical expressions and conditions are analyzed to determine subsets and their relationships.
Key concepts include:
- Definition and Implication: An asymmetric relation must satisfy certain conditions whereby pairs in the relation do not allow for symmetric counter-pairs. For instance, in order to deduce that A ⊆ B, we consider that (A ∩ C) ⊆ (B ∩ C)
- Mathematical Proofs: We demonstrate how certain properties hold under universally quantified statements and manipulate such expressions using logical equivalences. For example, if we have sets A, B, and C where for every x in set A, if x belongs to B implies it also belongs to C. We derive conditions to demonstrate relationships between the sets.
- Counting Relations: Different parts provide methods to count relations on a set S under various properties (symmetric, anti-symmetric, reflexive, irreflexive). For example, it is shown that for symmetric relations, the number of subsets from n² ordered pairs can be described mathematically, leading to generalized counting theorems.
This understanding and structuring lay the groundwork for more complex interactions between sets and relations, providing essential tools for the reader in discrete mathematics.
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Create a free accountThe requirement for an asymmetric relation states that if (a, b) is present in your relation, then (b, a) is not allowed in the relation. This does not mean that for every (a, b), you should have either (a, b) or (b, a) present in the relation. It is fine if none of them is there in your relation or not.
Detailed Explanation
An asymmetric relation is defined specifically by the condition that if one pair (a, b) is in the relation, the reverse pair (b, a) cannot be in the relation. This is different from being completely symmetric—it clarifies that both pairs can be absent, so it's possible to have relations where neither pair is present. It stresses the unidirectionality of the relationship represented by the pairs in the relation.
Examples & Analogies
Think of asymmetric relations like a one-way street. If you can go from A to B, you can't go back from B to A. Just like on a one-way street, you might have situations where neither direction is being traveled, which aligns with the rule that if (a, b) is there, (b, a) cannot be, but both could also be absent.
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Create a free accountAgain, let us club them into pairs of the form (i, j) and (j, i) where i and j are distinct. For such (i, j) and (j, i) pairs, I have three possibilities: I can include none of them, include (i, j) while excluding (j, i), or include (j, i) while excluding (i, j). However, we should not include both (i, j) and (j, i).
Detailed Explanation
When dealing with distinct pairs (i, j), we have specific choices to make. The relations can either include both pairs, include one, or include none. The key point is that including both pairs would violate the asymmetric property. Hence, we can make independent choices for distinct pairs but within the given restrictions, ensuring that we do not break the established asymmetry in the relation.
Examples & Analogies
Imagine a situation where you are deciding who to invite to a party. If you invite person A and they don’t invite you back (i.e., no mutual invite), you can either not invite them at all, or just invite A while deciding not to invite the reciprocal invite (you don’t expect to be invited back). This reflects the asymmetric nature—a one-sided invitation without reciprocation.
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Create a free accountTo determine the number of asymmetric relations, note that no diagonal elements are allowed, and for every distinct (i, j), you have three choices. If there are a total of n elements, the number of such distinct pairs is . Thus, the total number of asymmetric relations can be calculated as: 3^((n^2 - n)/2).
Detailed Explanation
The calculation begins with understanding that we cannot include any pairs where both elements are the same (diagonal elements). With 'n' elements, the number of non-diagonal pairs will be half of the total non-duplicated pairs, hence . Each pair gives rise to three choices, leading to the final formula for the count of asymmetric relations. It's essential to remember that these choices scale exponentially based on the number of elements.
Examples & Analogies
Think of assembling a playlist with strict rules about duplicates. If you have a set number of songs (n), you can choose to include the songs in various ways. Each song's presence or absence (in terms of being a pair or not) yields multiple playlist configurations—3 choices for each song's pair, making your selection exponentially complex as the number of songs increases.
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Key concepts
Core takeaways and short definitions to help you quickly recall the key ideas from this section.
- Asymmetric Relation:
If (a, b) is in the relation, (b, a) cannot be.
- Symmetric Relation:
If (a, b) is in the relation, (b, a) must also be.
- Anti-Symmetric Relation:
Allows (a, b) and (b, a) only if a = b.
- Reflexivity:
Requires (a, a) for all a in the set.
Examples
Step-by-step examples to apply the section's ideas and test your understanding.
The relation 'is a parent of' is asymmetric since if A is a parent of B, then B cannot be a parent of A.
The relation 'is equal to' is symmetric as if a = b, then b = a.
In a set of 3 elements, counting all possible asymmetric relations involves excluding pairs that contradict their definition.
Memory aids
Once upon a time, in a set kingdom, there lived pairs. One day, one pair went out, never to return home—because the rules said they couldn't be seen together!
Flash Cards
Glossary
Asymmetric Relation
A relation such that if (a, b) is in the relation, then (b, a) is not.
Symmetric Relation
A relation where if (a, b) is in the relation, then (b, a) is also in the relation.
Anti-Symmetric Relation
A relation where if both (a, b) and (b, a) are present, then a must equal b.
Reflexivity
A relation is reflexive if every element is related to itself, meaning (a, a) is in the relation for all a.
Diagonal Pair
Pairs of the form (a, a) for elements a in a set.