§ 3.3 Relations
What Is A Relation?
Definition
Let A and B be nonempty sets. A relation R from A to B is a subset of
the Cartesian product A × B. If R ⊆ A × B and if (a, b) ∈ R, we say
that a is related to b by R and we write a R b.
What Is A Relation?
Definition
Let A and B be nonempty sets. A relation R from A to B is a subset of
the Cartesian product A × B. If R ⊆ A × B and if (a, b) ∈ R, we say
that a is related to b by R and we write a R b.
Example
Let A = {1, 2, 3} and let B = {r, s}. Then R = {(1, r), (2, s), (3, r)}
is a relation from A to B.
Examples
Example
Let A and B be sets of real numbers. We define the relation R ‘equals’
from A to B as
aRb↔a=b
Examples
Example
Let A and B be sets of real numbers. We define the relation R ‘equals’
from A to B as
aRb↔a=b
Example
Let A = {1, 2, 3, 4}. Define the relation R (less than) as follows:
aRb↔a<b
Then
R = {(1, 2), (1, 3), (1, 4), (2, 3), (2, 4), (3, 4)}
Examples
Example
Let A be the set of positive integers. Define the following relation on A
a R b ↔ a|b
Then 4 R 12 but 56 R 7.
Examples
Example
Let A be the set of positive integers. Define the following relation on A
a R b ↔ a|b
Then 4 R 12 but 56 R 7.
Example
Let A be the set of all people in the world. We define the following
relation on A: a R b if and only if there is a sequence a0 , a1 , . . . an of
people such that a = a0 , b = an and ai−1 knows ai for i = 1, 2, . . . n.
Symmetry
Anyone remember what the symmetric property does for us?
Symmetry
Anyone remember what the symmetric property does for us?
Definition
A relation R is said to be symmetric if for (x, y) ∈ R then (y, x) ∈ R.
Symmetry
Anyone remember what the symmetric property does for us?
Definition
A relation R is said to be symmetric if for (x, y) ∈ R then (y, x) ∈ R.
Definition
A relation R is said to be asymmetric if for (x, y) ∈ R then (y, x) 6∈ R.
Symmetry
Anyone remember what the symmetric property does for us?
Definition
A relation R is said to be symmetric if for (x, y) ∈ R then (y, x) ∈ R.
Definition
A relation R is said to be asymmetric if for (x, y) ∈ R then (y, x) 6∈ R.
From our prior examples, can you think of any that were symmetric?
Asymmetric?
Transitivity
Does anyone remember what the transitive property does for us?
Transitivity
Does anyone remember what the transitive property does for us?
Definition
We say that a relation R is transitive if a R b and b R c implies a R c.
Transitivity
Does anyone remember what the transitive property does for us?
Definition
We say that a relation R is transitive if a R b and b R c implies a R c.
In terms of the Cartesian products, we have that the relation R is
transitive if for (x, y) ∈ R and (y, z) ∈ R that (x, z) ∈ R.
Transitivity
Does anyone remember what the transitive property does for us?
Definition
We say that a relation R is transitive if a R b and b R c implies a R c.
In terms of the Cartesian products, we have that the relation R is
transitive if for (x, y) ∈ R and (y, z) ∈ R that (x, z) ∈ R.
Any of our examples transitive?
The Reflexive Property
And how about the reflexive property - what does this property do for
us?
The Reflexive Property
And how about the reflexive property - what does this property do for
us?
Definition
A relation R is said to be reflexive if (x, x) ∈ R for all x in the domain.
The Reflexive Property
And how about the reflexive property - what does this property do for
us?
Definition
A relation R is said to be reflexive if (x, x) ∈ R for all x in the domain.
Definition
A relation R that is asymmetric, transitive and reflexive is called a
partial order.
The Reflexive Property
And how about the reflexive property - what does this property do for
us?
Definition
A relation R is said to be reflexive if (x, x) ∈ R for all x in the domain.
Definition
A relation R that is asymmetric, transitive and reflexive is called a
partial order.
Definition
A relation R that is symmetric, transitive and reflexive is called an
equivalence relation.
Inverses
Definition
The inverse of a relation R, denoted R−1 is the set
{(y, x) | (x, y) ∈ R}
Inverses
Definition
The inverse of a relation R, denoted R−1 is the set
{(y, x) | (x, y) ∈ R}
Example
For the relation we defined on the set A = {1, 2, 3, 4} where
aRb↔a<b
we have
a R−1 b ↔ a ≥ b
Visual Representations - Digraphs
Definition
A digraph, or directed graph, is a graph, or set of nodes connected by
edges, where the edges have a direction associated with them.
Visual Representations - Digraphs
Definition
A digraph, or directed graph, is a graph, or set of nodes connected by
edges, where the edges have a direction associated with them.
We can use digraphs to give visual representations of relations.
Visual Representations - Digraphs
Example
Give the relation represented by the digraph
Visual Representations - Digraphs
Example
Give the relation represented by the digraph
R = {(1, 1), (1, 2), (1, 3), (2, 3)}
Visual Representations - Digraphs
Example
Give a visual representation for the relation R on A = {1, 2, 3, 4}
where
R = {(1, 2), (2, 3), (2, 4), (3, 3), (4, 2)}
Visual Representations - Digraphs
Example
Give a visual representation for the relation R on A = {1, 2, 3, 4}
where
R = {(1, 2), (2, 3), (2, 4), (3, 3), (4, 2)}
Visual Representations - Digraphs
Example
Give a visual representation for the relation R on A = {1, 2, 3, 4}
where
R = {(1, 1), (1, 2), (2, 2), (2, 3), (2, 4), (3, 4), (4, 1)}
Visual Representations - Digraphs
Example
Give a visual representation for the relation R on A = {1, 2, 3, 4}
where
R = {(1, 1), (1, 2), (2, 2), (2, 3), (2, 4), (3, 4), (4, 1)}