18 Questions
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.
If (a,b) belongs to the relation R, we say 'a is R-related to b'.
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.
Set inclusion ⊆ is a relation on any collection of sets where A is a subset of B if every member of A is a member of B.
Explain what it means for set A to be a subset of set B.
Set A is a subset of set B if every member of A is also a member of B.
What is the range of a relation R?
The range of a relation R is the set of all second elements of the ordered pairs in R.
Is the relation R₉ = {(2,1),(1,3)} transitive? Why or why not?
No
Define an asymmetric relation and provide an example.
An asymmetric relation is a relation where for all elements (a,b) in R, (b,a) is not in R. Example: R₁₁ = {(1,2),(1,3),(2,3)}
What is the definition of an irreflexive relation?
For all a in A, (a,a) is not in R
Explain the concept of an equivalence relation.
An equivalence relation is reflexive, symmetric, and transitive. It partitions the set into equivalent classes.
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?
No
Is R₅ = Ø an equivalence relation? Why or why not?
No
Give an example of a reflexive relation on the set A = {1,2,3,4}.
R₁ = {(1,1),(1,2),(2,1),(2,2),(2,3),(3,3),(4,4)}
Define an irreflexive relation and provide an example on the set A = {1,2,3,4}.
An irreflexive relation does not contain any ordered pairs of the form (a,a) for any element a ∈ A. Example: R₃ = {(1,2),(2,1)}
Explain what a symmetric relation is and give an example of a symmetric relation.
A symmetric relation is one where if (a,b) is in the relation, then (b,a) is also in the relation. Example: R₇ = {(1,1),(2,1)}
What is an antisymmetric relation and provide an example of an antisymmetric relation?
An antisymmetric relation is one where if (a,b) is in the relation and (b,a) is also in the relation, then a must equal b. Example: R₇ = {(1,1),(2,1)}
Define a transitive relation. Give an example of a transitive relation on the set A = {1,2,3,4}.
A transitive relation is one where if (a,b) and (b,c) are in the relation, then (a,c) must also be in the relation. Example: R₈ = {(2,1),(3,1),(3,2),(4,4)}
Is the relation R₂ = {(1,1),(1,2),(2,1),(2,2),(3,1),(4,4)} reflexive? Why or why not?
No, R₂ is not reflexive because the ordered pair (3,3) is not in R₂.
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.
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