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Questions and Answers
Define a binary relation.
Define a binary relation.
A binary relation from set A to set B is a subset of the Cartesian product of A and B.
Explain what it means for an element 'a' to be 'R-related' to an element 'b' in a relation R.
Explain what it means for an element 'a' to be 'R-related' to an element 'b' in a relation R.
If (a,b) belongs to the relation R, we say 'a is R-related to b'.
What is the domain of a relation R?
What is the domain of a relation R?
The domain of a relation R from set A to set B is the set of all first elements of the ordered pairs in R.
Define set inclusion as a relation on any collection of sets.
Define set inclusion as a relation on any collection of sets.
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Explain what it means for set A to be a subset of set B.
Explain what it means for set A to be a subset of set B.
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What is the range of a relation R?
What is the range of a relation R?
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Is the relation R₉ = {(2,1),(1,3)} transitive? Why or why not?
Is the relation R₉ = {(2,1),(1,3)} transitive? Why or why not?
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Define an asymmetric relation and provide an example.
Define an asymmetric relation and provide an example.
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What is the definition of an irreflexive relation?
What is the definition of an irreflexive relation?
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Explain the concept of an equivalence relation.
Explain the concept of an equivalence relation.
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Is R₄ = {(0,0),(0,1),(0,2),(1,0),(1,1),(1,2),(2,0),(2,2),(3,3)} an equivalence relation? Why or why not?
Is R₄ = {(0,0),(0,1),(0,2),(1,0),(1,1),(1,2),(2,0),(2,2),(3,3)} an equivalence relation? Why or why not?
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Is R₅ = Ø an equivalence relation? Why or why not?
Is R₅ = Ø an equivalence relation? Why or why not?
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Give an example of a reflexive relation on the set A = {1,2,3,4}.
Give an example of a reflexive relation on the set A = {1,2,3,4}.
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Define an irreflexive relation and provide an example on the set A = {1,2,3,4}.
Define an irreflexive relation and provide an example on the set A = {1,2,3,4}.
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Explain what a symmetric relation is and give an example of a symmetric relation.
Explain what a symmetric relation is and give an example of a symmetric relation.
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What is an antisymmetric relation and provide an example of an antisymmetric relation?
What is an antisymmetric relation and provide an example of an antisymmetric relation?
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Define a transitive relation. Give an example of a transitive relation on the set A = {1,2,3,4}.
Define a transitive relation. Give an example of a transitive relation on the set A = {1,2,3,4}.
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Is the relation R₂ = {(1,1),(1,2),(2,1),(2,2),(3,1),(4,4)} reflexive? Why or why not?
Is the relation R₂ = {(1,1),(1,2),(2,1),(2,2),(3,1),(4,4)} reflexive? Why or why not?
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Study Notes
Binary Relations
- A binary relation on a set is a subset of the Cartesian product of that set with itself, representing associations between pairs of elements.
R-Related Elements
- An element 'a' is said to be 'R-related' to an element 'b' if the ordered pair (a, b) exists in the relation R.
Domain of a Relation
- The domain of a relation R is the set of all first elements (or inputs) from the ordered pairs in R.
Set Inclusion
- Set inclusion is defined as a relation where for any sets A and B, A is included in B if every element of A is also an element of B.
Subset Definition
- Set A is a subset of set B if every element of A is also in B, denoted as A ⊆ B.
Range of a Relation
- The range of a relation R includes all second elements (or outputs) from the ordered pairs in R.
Transitivity of R₉
- The relation R₉ = {(2,1),(1,3)} is not transitive because there is no pair (2,3) to satisfy the transitive property - if (a,b) and (b,c) are in R, then (a,c) must also be in R.
Asymmetric Relation
- An asymmetric relation is one where if (a, b) is in the relation, then (b, a) cannot be in the relation.
- Example: The relation "is less than" on the set of real numbers.
Irreflexive Relation
- An irreflexive relation does not allow any element to be related to itself; no pair (a, a) is in the relation.
Equivalence Relation
- An equivalence relation satisfies three properties: reflexivity, symmetry, and transitivity. These properties create equivalence classes within the set.
Equivalence Status of R₄
- R₄ = {(0,0),(0,1),(0,2),(1,0),(1,1),(1,2),(2,0),(2,2),(3,3)} is not an equivalence relation because it is not symmetric (e.g., (0,1) is in R₄ but (1,0) is not).
Equivalence Status of R₅
- R₅ = Ø is vacuously an equivalence relation since it does not violate any of the properties.
Reflexive Relation Example
- A reflexive relation on the set A = {1,2,3,4} could be R₆ = {(1,1), (2,2), (3,3), (4,4)}, where every element relates to itself.
Irreflexive Relation Example
- An example of an irreflexive relation on A = {1,2,3,4} is R₇ = {(1,2), (2,3), (3,4)}, where no element is related to itself.
Symmetric Relation
- A symmetric relation is one in which if (a, b) is in R, then (b, a) is also in R.
- Example: R₈ = {(1,2), (2,1)}.
Antisymmetric Relation
- An antisymmetric relation states that if (a, b) and (b, a) are both in the relation, then a must equal b.
- Example: The relation "is less than or equal to" on real numbers.
Transitive Relation
- A transitive relation holds that if (a, b) and (b, c) are in the relation, then (a, c) must also be in the relation.
Transitive Relation Example
- An example of a transitive relation on the set A = {1,2,3,4} is R₉ = {(1,2), (2,3), (1,3)}.
Reflexivity of R₂
- The relation R₂ = {(1,1),(1,2),(2,1),(2,2),(3,1),(4,4)} is not reflexive since not every element in the set {1,2,3,4} is related to itself; for instance, (3,3) is missing.
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Description
Learn about binary relations between sets A and B, where a relation from A to B is a subset of A×B consisting of ordered pairs. Understand how each pair connects elements from A to B, and how to determine if elements are related within the relation R.
