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1.1. Definition of a Relation
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Today we'll dive into the definition of a relation, which is a subset of the Cartesian product of two sets. Can anyone tell me what they think a Cartesian product is?
Isn't that when you combine all pairs of elements from two sets?
Exactly! If we have sets A and B, the Cartesian product A × B consists of all possible ordered pairs (a, b) where 'a' is from A and 'b' is from B. A relation is simply a special selection of some of those pairs.
Can you give an example?
Of course! If A = {1, 2, 3} and B = {a, b, c}, one possible relation R could be R = {(1, a), (2, b), (3, c)}.
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Now, let's discuss the types of relations. Who can explain what a reflexive relation is?
I think it's when every element relates to itself, right? Like (x, x) for every x?
That's correct! An example of a reflexive relation is R = {(1, 1), (2, 2), (3, 3)} if A = {1, 2, 3}. How about symmetric relations?
A relation is symmetric if for every pair (a, b), (b, a) is also in the set.
Great job! Now, if we have R = {(1, 2), (2, 1)}, that's symmetric. Let's not forget about transitive relations. Who can give me a definition?
It's when if (a, b) and (b, c) are both in R, then (a, c) must also be in R.
Exactly! If R = {(1, 2), (2, 3), (1, 3)}, this relation is transitive. There are also anti-symmetric and equivalence relations to explore!
Overview
Short Summary
A relation is a subset of the Cartesian product of two sets, consisting of ordered pairs.
Medium Summary
This section covers the definition of relation as a subset of the Cartesian product of two sets, explores different types of relations including reflexive, symmetric, transitive, anti-symmetric, and equivalence relations, and illustrates these concepts with examples.
Detailed Summary
Definition of a Relation
In mathematics, a relation between two sets, denoted as A and B, is defined as a subset of the Cartesian product A × B. This means that a relation is formed by a set of ordered pairs (a, b), where 'a' is an element from set A and 'b' is an element from set B.
Types of Relations:
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Reflexive Relation: A relation R on a set A is reflexive if (x, x) is in R for every element x in A.
- Example: Let A = {1, 2, 3}. A reflexive relation on A is R = {(1, 1), (2, 2), (3, 3)}.
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Symmetric Relation: A relation R is symmetric if for every pair (a, b) in R, the pair (b, a) is also in R.
- Example: If R = {(1, 2), (2, 1)}, the relation is symmetric.
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Transitive Relation: A relation R is transitive if whenever (a, b) is in R and (b, c) is in R, it follows that (a, c) is also in R.
- Example: If R = {(1, 2), (2, 3), (1, 3)}, then it is transitive.
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Anti-symmetric Relation: A relation R is anti-symmetric if for every (a, b) in R and (b, a) in R, it must be the case that a = b.
- Example: If R = {(1, 2), (2, 1)}, the relation is not anti-symmetric since 1 ≠ 2.
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Equivalence Relation: A relation is an equivalence relation if it is reflexive, symmetric, and transitive.
- Example: The relation R = {(1, 1), (2, 2), (1, 2), (2, 1)} is an equivalence relation.
Understanding these types is crucial as they form the basis for many advanced mathematical concepts and operations.
Audio Book
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Create a free accountA relation between two sets 𝐴 and 𝐵 is a subset of the cartesian product 𝐴×𝐵. This means that a relation is a set of ordered pairs, where the first element is from set 𝐴 and the second element is from set 𝐵.
Detailed Explanation
A relation connects two sets by associating elements from one set (set A) with elements from another set (set B). The Cartesian product 𝐴×𝐵 generates all possible pairs (x, y) where x is from set A and y is from set B. A relation is a specific selection of some of these pairs, which form a subset of the Cartesian product.
Examples & Analogies
Consider a school where set A represents students and set B represents classes. A relation could represent which student is enrolled in which class, such as { (Alice, Math), (Bob, Science) }. This signifies that Alice is in Math, and Bob is in Science.
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Create a free accountLet 𝐴 = {1,2,3} and 𝐵 = {𝑎,𝑏,𝑐}. A relation from 𝐴 to 𝐵 is a subset of 𝐴×𝐵, for example: 𝑅 = {(1,𝑎),(2,𝑏),(3,𝑐)}.
Detailed Explanation
In this example, we have set A containing the numbers 1, 2, and 3, while set B contains the letters a, b, and c. The relation R consists of ordered pairs where each number from set A is paired with a letter from set B. This clearly shows how elements from both sets can be related in a structured way.
Examples & Analogies
Imagine a fruit shop where set A consists of fruit types {Apple, Banana, Cherry} and set B consists of colors {Red, Yellow, Pink}. A relation could be {(Apple, Red), (Banana, Yellow), (Cherry, Pink)}, which defines the color of each fruit at the shop.
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Key Concepts
Core takeaways and short definitions to help you quickly recall the key ideas from this section.
Relation: A subset of ordered pairs from the Cartesian product of two sets.
Reflexive Relation: A relation where every element is related to itself.
Symmetric Relation: A symmetric relation (mutual inclusion of ordered pairs).
Transitive Relation: A relation maintaining a certain chain of relationships.
Anti-symmetric Relation: A relation where one direction implies equality.
Equivalence Relation: A relation that is reflexive, symmetric, and transitive.
Examples
Step-by-step examples to apply the section's ideas and test your understanding.
Let A = {1, 2, 3} and B = {a, b, c}. A possible relation R could be R = {(1, a), (2, b), (3, c)}.
If R = {(1, 2), (2, 3), (1, 3)}, R is transitive.
If R = {(1, 2), (2, 1)}, R is symmetric, while R = {(1, 2), (2, 3)} is transitive.
Memory Aids
Interactive tools to help you remember key concepts
Stories
Flash Cards
Glossary
Relation
A subset of the Cartesian product of two sets, consisting of ordered pairs.
Reflexive Relation
A relation R on a set A is reflexive if every element relates to itself, i.e., (x, x) ∈ R for every x ∈ A.
Symmetric Relation
A relation R is symmetric if whenever (a, b) ∈ R, then (b, a) ∈ R.
Transitive Relation
A relation R is transitive if whenever (a, b) ∈ R and (b, c) ∈ R, then (a, c) ∈ R.
Antisymmetric Relation
A relation R is anti-symmetric if, whenever (a, b) ∈ R and (b, a) ∈ R, then a = b.
Equivalence Relation
A relation that is reflexive, symmetric, and transitive.