Question 3
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Understanding Symmetric Relations
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Let's start with symmetric relations. Who can tell me what it means for a relation to be symmetric?
Isn't it that if you have a pair (i, j) in the relation, you must also have (j, i)?
Exactly! And for how many pairs can we do this for a set of n elements?
I think it's based on the upper triangular pairs in an n x n matrix?
Correct! The number of symmetric relations is given by 2 raised to the power of n(n + 1)/2. Can anyone summarize why we can select from just these pairs?
Because including both (i, j) and (j, i) would violate symmetry if they are different, right?
That's right! To recap, symmetric relations require you to include corresponding pairs; the total arrangements are based on the combinations of upper triangular pairs.
Exploring Anti-symmetric Relations
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Now, let's shift to anti-symmetric relations. Who can describe this property?
I believe it's when if both (i, j) and (j, i) are present in the relation, `i` must equal `j`?
Correct! So if `i` is not equal to `j`, having both pairs is a violation. Can someone explain how we calculate the number of such relations?
We have options for diagonal pairs, and for non-diagonal pairs, we have to split them as either one can be selected, but not both.
Exactly! Thus the total becomes 2^n times 3^(n(n - 1)/2). Can anyone summarize what happens with diagonal elements?
Diagonal elements can be included or excluded without restrictions, meaning we have flexibility there.
Great recap! Understanding these combinations helps us navigate counting effectively.
Recapping Asymmetric and Irreflexive Relations
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Let's discuss asymmetric relations. How is this property distinct from anti-symmetric?
Asymmetric means if (i, j) is present, then (j, i) cannot be present at all, right?
Exactly! And can we include diagonal elements?
No, including them would violate the condition since they are self-referential.
Exactly! Now, if we are to find the total number of asymmetric relations, how would we go about that?
By excluding diagonal elements and allowing any non-diagonal tuples as long as none of their opposites are included?
That's the essence! Now let’s talk about irreflexive relations. What defines them?
An irreflexive relation has no diagonal pairs included at all! They must be absent.
Spot on! By excluding diagonal elements, the remaining pairs can be selected freely. This interconnectedness of these properties is crucial for counting.
Combining Properties
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Let's analyze relations that are both symmetric and anti-symmetric. What can we infer?
Well, the only way to satisfy both properties is to include all diagonal pairs and omit others!
Correct! If we include off-diagonal pairs, symmetry fails. How many relations meet both conditions?
"I think it’s only the identity relation with all diagonal pairs included, so 1 relation!
Introduction & Overview
Read summaries of the section's main ideas at different levels of detail.
Quick Overview
Standard
This section elaborates on how to determine the number of relations that can exist within a set, based on properties such as symmetry, anti-symmetry, reflexivity, and irreflexivity. It discusses how to count these relations through logical reasoning and combinatorial methods.
Detailed
Counting Relations on a Set
In this section, we delve into the properties of relations on a set of size n where n >= 1. The properties discussed include symmetric, anti-symmetric, asymmetric, irreflexive, reflexive, and their combinations.
Key Concepts:
-
Symmetric Relations: A relation is symmetric if for every pair
(i, j), whenever(i, j)is in the relation,(j, i)is also in the relation. The total number of symmetric relations that can be formed fromnelements is calculated by selecting ordered pairs from the upper triangular sorted pairs of ann x nmatrix, resulting in2^(n(n+1)/2)possible symmetric relations. -
Anti-symmetric Relations: A relation is anti-symmetric if whenever
(i, j)and(j, i)are present, thenimust equalj. For anti-symmetric relations, we separately consider diagonal elements and the pairs(i, j)for distinctiandj, leading to2^n * 3^(n^2 - n)/2possible configurations. -
Asymmetric Relations: A relation is asymmetric if for any
(i, j)in the relation,(j, i)cannot be present. Thus all diagonal elements are excluded, and the count yields3^(n^2 - n)/2possible asymmetric relations. -
Irreflexive Relations: For a relation to be irreflexive, none of the diagonal elements
(a, a)can be included, leading to2^(n^2 - n)possible relations. - Reflexive and Symmetric Relations: For a combination of reflexive and symmetric relations, all diagonal pairs must be included, rendering exactly one such relation.
- Combinations: The section also discusses relationships that are neither reflexive nor irreflexive and how these overlap in terms of counting distinct relations.
This exploration of set relations emphasizes not just the definitions but also how to use combinatorial reasoning to count them effectively.
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Introduction to Relations
Chapter 1 of 10
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Chapter Content
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.
Detailed Explanation
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.
Examples & Analogies
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.
Counting Symmetric Relations
Chapter 2 of 10
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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.
Detailed Explanation
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.
Examples & Analogies
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.
Subset Selection for Symmetry
Chapter 3 of 10
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Chapter Content
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.
Detailed Explanation
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.
Examples & Analogies
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.
Constructing Symmetric Relations
Chapter 4 of 10
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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.
Detailed Explanation
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.
Examples & Analogies
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.
Counting Anti-Symmetric Relations
Chapter 5 of 10
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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...
Detailed Explanation
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.
Examples & Analogies
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.
Determining Relations Based on Conditions
Chapter 6 of 10
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So, let us focus on the non-diagonal ordered pairs of the form (i, j) and (j, i) where i and j are distinct...
Detailed Explanation
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.
Examples & Analogies
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.
Identifying Asymmetric Relations
Chapter 7 of 10
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Chapter Content
In part C of the third question, you are supposed to find out the number of relations which are asymmetric...
Detailed Explanation
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.
Examples & Analogies
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.
Understanding Irreflexive Relations
Chapter 8 of 10
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Chapter Content
In part d, I am interested to find out how many relations I can form which are irreflexive...
Detailed Explanation
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.
Examples & Analogies
Imagine a sports team rule where players cannot score against themselves — this ensures that all interactions are with others, not with their own actions.
Reflexive and Symmetric Relations
Chapter 9 of 10
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Chapter Content
Part e, I now want to find the number of relations which are simultaneously reflexive as well as symmetric...
Detailed Explanation
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.
Examples & Analogies
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).
Final Considerations
Chapter 10 of 10
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Chapter Content
The last part of the question, I have to find out the number of relations, which are neither reflexive nor irreflexive...
Detailed Explanation
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.
Examples & Analogies
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).
Key Concepts
-
Symmetric Relations: A relation is symmetric if for every pair
(i, j), whenever(i, j)is in the relation,(j, i)is also in the relation. The total number of symmetric relations that can be formed fromnelements is calculated by selecting ordered pairs from the upper triangular sorted pairs of ann x nmatrix, resulting in2^(n(n+1)/2)possible symmetric relations. -
Anti-symmetric Relations: A relation is anti-symmetric if whenever
(i, j)and(j, i)are present, thenimust equalj. For anti-symmetric relations, we separately consider diagonal elements and the pairs(i, j)for distinctiandj, leading to2^n * 3^(n^2 - n)/2possible configurations. -
Asymmetric Relations: A relation is asymmetric if for any
(i, j)in the relation,(j, i)cannot be present. Thus all diagonal elements are excluded, and the count yields3^(n^2 - n)/2possible asymmetric relations. -
Irreflexive Relations: For a relation to be irreflexive, none of the diagonal elements
(a, a)can be included, leading to2^(n^2 - n)possible relations. -
Reflexive and Symmetric Relations: For a combination of reflexive and symmetric relations, all diagonal pairs must be included, rendering exactly one such relation.
-
Combinations: The section also discusses relationships that are neither reflexive nor irreflexive and how these overlap in terms of counting distinct relations.
-
This exploration of set relations emphasizes not just the definitions but also how to use combinatorial reasoning to count them effectively.
Examples & Applications
An example of a symmetric relation is { (1, 2), (2, 1), (3, 3) } where (1, 2) includes (2, 1).
An example of an irreflexive relation is { (1, 2), (2, 3) } which does not include any pairs like (1, 1) or (2, 2).
Memory Aids
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Rhymes
For symmetric ties, pair the highs, (i,j) and (j,i) must arise.
Stories
In a magical set forest, all pairs twirled around. A pair of friends named i and j danced together, but if i twirled with j, j must mimic, twirling back.
Memory Tools
Remember: ABC - A pair must go both ways for symmetry; B in anti-symmetry means no pairs except self; C in irreflexive means 'no self-love'.
Acronyms
SIR - Symmetric, Irreflexive; A means Anti-symmetric relations cannot exist for both pairs being different.
Flash Cards
Glossary
- Symmetric Relation
A relation where if (i, j) is included, then (j, i) must also be included.
- Antisymmetric Relation
A relation such that if both (i, j) and (j, i) are present, then i must equal j.
- Asymmetric Relation
A relation where if (i, j) is included, (j, i) cannot be included.
- Irreflexive Relation
A relation that does not include any diagonal pairs (a, a).
- Reflexive Relation
A relation that includes all pairs (a, a) for every a in the set.
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