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Questions and Answers
Given the relation R defined over the set A = {1, 2, 3, 4, 5...} where (a, b) ∈ R if a - b is divisible by 3, which of the following pairs is NOT in R?
Given the relation R defined over the set A = {1, 2, 3, 4, 5...} where (a, b) ∈ R if a - b is divisible by 3, which of the following pairs is NOT in R?
Consider a relation S on the set of integers where (a, b) ∈ S if a - b is divisible by 5. Is S transitive?
Consider a relation S on the set of integers where (a, b) ∈ S if a - b is divisible by 5. Is S transitive?
Which of the following statements is TRUE about an equivalence relation?
Which of the following statements is TRUE about an equivalence relation?
Consider the relation T on the set of all real numbers where (a, b) ∈ T if a = b. Is the relation T transitive?
Consider the relation T on the set of all real numbers where (a, b) ∈ T if a = b. Is the relation T transitive?
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Given the relation R defined over the set of all integers where (a, b) ∈ R if a + b is even, is the relation R symmetric?
Given the relation R defined over the set of all integers where (a, b) ∈ R if a + b is even, is the relation R symmetric?
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Given a relation R, what must be true for it to be considered reflexive?
Given a relation R, what must be true for it to be considered reflexive?
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What is the range of the relation R = {(1, 1), (2, 4), (3, 9)}?
What is the range of the relation R = {(1, 1), (2, 4), (3, 9)}?
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If a relation R is symmetric, what can we conclude about the pairs (a, b) and (b, a)?
If a relation R is symmetric, what can we conclude about the pairs (a, b) and (b, a)?
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Which of the following relations are NOT transitive? (Select all that apply)
Which of the following relations are NOT transitive? (Select all that apply)
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Consider relation R = {(1, 1), (1, 2), (2, 1), (2, 2), (1, 3)}. What is the codomain of this relation?
Consider relation R = {(1, 1), (1, 2), (2, 1), (2, 2), (1, 3)}. What is the codomain of this relation?
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What is the MOST defining characteristic of an Equivalence Relation?
What is the MOST defining characteristic of an Equivalence Relation?
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If the function a = x² is used to define a relation, why wouldn't this relation be reflexive?
If the function a = x² is used to define a relation, why wouldn't this relation be reflexive?
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Consider a relation where Set A = {1, 2, 3} and Set B = {2, 3, 4}. Which of the following relations would be symmetric? (Select all that apply)
Consider a relation where Set A = {1, 2, 3} and Set B = {2, 3, 4}. Which of the following relations would be symmetric? (Select all that apply)
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Flashcards
Reflexivity
Reflexivity
A relation R is reflexive if every element is related to itself.
Symmetry
Symmetry
A relation R is symmetric if (a, b) implies (b, a) for any elements a and b.
Transitivity
Transitivity
A relation R is transitive if (a, b) and (b, c) imply (a, c).
Equivalence Relation
Equivalence Relation
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Divisibility
Divisibility
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Relation
Relation
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Cartesian Product
Cartesian Product
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Reflexive Relation
Reflexive Relation
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Symmetric Relation
Symmetric Relation
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Transitive Relation
Transitive Relation
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Domain of a Relation
Domain of a Relation
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Range of a Relation
Range of a Relation
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Study Notes
What is a Relation?
- A relation exists when two sets are defined with a function between them.
- The Cartesian product of the sets determines if the relation is valid.
- Three key concepts in relations are reflexivity, symmetry, and transitivity.
Types of Relations
- A relation is reflexive if, for every element x in set A, the pair (x, x) is in the relation.
- A relation is symmetric if, for every (a, b) in the relation, (b, a) must also be in the relation.
- A relation is transitive if, for every (a, b) and (b, c) in the relation, (a, c) must also be in the relation.
- An equivalence relation satisfies all three properties (reflexive, symmetric, transitive).
Domain, Range, and Codomain
- The domain of a relation is the set of all "first" elements of the pairs in the relation.
- The range of a relation is the set of all "second" elements of the pairs in the relation.
- The codomain of a relation is the set of all possible "second" elements; usually the right-hand set in the relation.
Examples & Properties of Relations
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Example 1: Set A = {1, 2, 3, 4, 5, 6, 7}, Set B = {1, 2, 3, 4, 5, 6, 7, 8, 9,...}. Function: a = x².
- Relation R: (1, 1), (2, 4), (3, 9), ...
- Domain of R: {1, 2, 3, 4, 5, 6, 7}
- Range of R: {1, 4, 9, 16, 25, 36, ...}
- Codomain of R: All natural numbers (including 0).
- R is not reflexive, symmetric, or transitive.
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Example 2: Set A = {1, 2}, Set B = {1, 2, 3, 4}. Relation R = {(1, 1), (1, 2), (2, 1), (2, 3), (2, 4)}.
- R is reflexive (1, 1) and (2, 2) are present but not for all elements
- R is symmetric (1, 2) and (2, 1) are in R.
- R is not transitive (1, 2) and (2, 3) are in R, but (1, 3) is not.
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Example 3: Set A = {1, 2, 3, 4}, Relation R based on a = x².
- R is not reflexive, symmetric, or transitive.
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Example 4: Set A = {1, 2, 3, 4, 5,...}. Relation R: (a, b) where (a - b) is divisible by 3.
- R is reflexive (a - a = 0 is divisible by 3).
- R is symmetric (if a - b is divisible by 3 then b - a is also divisible by 3).
- R is transitive (if a - b and b - c are divisible by 3 then a - c is divisible by 3).
Key Ideas
- Divisibility is crucial in determining transitive relations.
Applying the Concepts
- To identify an equivalence relation:
- Check for reflexivity.
- Check for symmetry.
- Check for transitivity.
- If all three conditions are met, it's an equivalence relation.
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Description
This quiz focuses on the fundamental concepts of relations in mathematics, including definitions and properties such as reflexive, symmetric, and transitive relations. Additionally, it explores the concepts of domain, range, and codomain. Test your understanding of these crucial mathematical principles.
