Relations
Ordered Pairs
Given a non-empty set S, an ordered pair of elements of S, denoted
by (a, b), consists of a pair of elements of S ( a and b, which need
not be distinct) for which one is considered the "first" element and
the other the "second" element.
Thus, as subsets {a, b} = {b, a} but as ordered pairs (a, b) ≠ (b, a).
Also note that while (a, a) is a legal ordered pair, {a,a} is not a set.
Two ordered pairs (a, b) and (c, d) are equal iff a = c and b = d.
Ordered Pairs
We point out that this is only a naïve and not a real definition. As
we shall see, the concept of order (used above as first and second)
is itself defined in terms of ordered pairs ... so we really have set
up a bit of circular reasoning, but this can be fixed.
The real definition of an ordered pair is:
(a, b) = {{a}, {a,b}}.
To see why we would prefer to deal with ordered pairs naïvely
consider the proof of the statement:
(a, b) = (c, d) iff a = c and b = d
using the real definition.
(a, b) = (c, d) iff a = c and b = d
Pf: The necessity of the condition is easy to see.
(a, b) = {{a}, {a,b}} = {{c}, {c,d}} = (c,d).
To prove the sufficiency, assume (a,b) = (c,d).
Thus, {{a}, {a,b}} = {{c}, {c,d}}.
Now, either {a} = {c} or {a} = {c,d}.
Case I: {a} = {c}
It follows that a = c. We also have that {a,b} = {c, b} = {c,d}.
Now b = d follows from the second equality.
Case II: {a} = {c,d}
This can only be true if c = d, and it would follow that a = c = d.
We would also have {a,b} = {c}, which requires that a = b = c.
So, we have a = b = c = d and may conclude that a = c and b = d.
Cartesian Products
Given sets A and B, we define the Cartesian product of A and B,
denoted by A×B, by
A×B = {(a,b) | a ∈ A ∧ b ∈ B }.
(Note that the set S in the "definition" of ordered pairs would be
A∪B in this case.)
Example: Let A = {1, 2, 3} and B = {2, 4}, then
A×B = {(1,2), (1,4), (2,2), (2,4), (3,2), (3,4)} while
B×A = {(2,1), (2,2), (2,3), (4,1), (4,2),(4,3)}.
so we see that in general A×B ≠ B×A.
Cartesian Products
Two observations:
(a, b) ∉ A × B ≡ a ∉ A ∨ b ∉ B.
Even though A × B is a set, in proofs involving A × B it is not a
good idea to write " let x ∈ A × B ... " because you would then
have to continue to remember that x is an ordered pair and not a
single element. The better way is to write
" let (x,y) ∈ A × B ... "
since this makes the ordered pair nature of the A × B elements
explicit.
Relations
Let A and B be sets. A relation from A to B is a subset of A×B.
If R is a relation from A to B and (a,b) ∈ R, we often write a R b
and say "a is R-related to b" or just "a is related to b" if R is
understood.
When A = B, we refer to a relation on A.
Examples:
Let A = {1, 2, 3, 4} and define the relation "|" (= R) on A to be
integer division.
Then, | = {(1,1), (1,2), (1,3), (1,4), (2,2), (2,4), (3,3), (4,4)}.
It is more common to write, 1|2, 2|4, 3|3 etc. instead of (1,2) ∈ |,
(2,4) ∈ |, (3,3) ∈ |, etc.
Examples of Relations
Again let A = {1,2,3,4} and define the relation ">" on A by:
> = {(2,1), (3,2), (3,1), (4,3), (4,2), (4,1)}
We see that ">" has its usual meaning 3 > 2, 4 > 3, 4 > 1 etc.
Let A = {1, 4, 9, 16} and B = {1, 2, 3, 4} then the relation R from
A to B given by:
R = {(1,1), (4,2), (9,3), (16,4)}
can be interpreted as "is the square of", i.e., 4 is the square of 2,
16 is the square of 4, etc.
Let A = {1, 2, 3} and B = {1, 3, 6, 7}, we can define a relation R
from A to B by:
R = {(x,y) | y = 2x + 1}
then R = {(1,3), (3,7)}
Relations
Relations are very general.
The mathematical term "relation" is in fact more general than the
corresponding English word "relation". In English, when you use
the phrase "a is related to b" you mean that there is some "link"
between a and b. No such "link" need be present in a mathematical
relation.
To say that a and b are not R-related we either use the notation
(a,b) ∉ R or a R
/ b.
Domains and Ranges
The domain of a relation R from A to B is the set:
Dom(R) = {x ∈ A | ∃ y ∈ B such that x R y }
= {x ∈ A | ∃ y ∈ B such that (x,y) ∈ R}
= {x ∈ A | x is a first coordinate of some pair in R}.
The range of a relation R from A to B is the set:
Rng(R) = {y ∈ B | ∃ x ∈ A such that x R y }
= {y ∈ B | ∃ x ∈ A such that (x,y) ∈ R }
= {y ∈ B | y is a second coordinate of some pair in R}.
Clearly Dom(R) ⊆ A and Rng(R) ⊆ B.
Graphs of Relations
In many situations it is convenient to have a pictorial representation
of a relation.
The Cartesian graph of a relation is obtained by setting up a
coordinate axis system in which the "positions" on the X-axis
correspond to the elements of A, and the "positions" on the Y-axis
correspond to the elements of B. The ordered pairs of R are
represented as points with respect to these two axes.
For general sets A and B, the spacing and ordering of the positions
on the X- and Y-axes are arbitrary. Thus, the "picture" that you get
is not unique and does not carry much information.
A Cartesian Graph
Let A = {1, 4, 9, 16, 18} and B = {1, 2, 3, 4, 5} and the relation R
from A to B given by R = {(1,1), (4,2), (9,3), (16,4)}. A Cartesian
graph of R may look like:
5
Rng(R)
4
3
2
Dom(R)
1
1
4
9
16
18
Digraphs
With relations on a set, there is a second way to provide a pictorial
representation. These are known as digraphs (directed graphs).
Each element of the set A is represented by a point (vertex) and for
each (a,b) in R, an arrow is drawn from the point labeled a to the
point labeled b (called an arc or directed edge).
Example: Let A = {1, 2, 3, 4} and R = {(1,1), (2,3), (3,2), (4,2)}.
2
1
3
4
Dom(R) = A
= vertices with out arcs
Rng(R) = {1,2,3}
= vertices with in arcs
Properties of Relations on a Set
A relation R on a set A may have any of the following special
properties.
A relation R on A is reflexive iff ∀ a ∈ A, (a,a) ∈ R.
(That is, the relation R contains the identity relation IA).
A relation R on A is symmetric iff ∀ a,b ∈ A
(a,b) ∈ R iff (b,a) ∈ R.
A relation R on A is antisymmetric iff ∀ a,b ∈ A
if (a,b) ∈ R and (b,a) ∈ R then a = b.
A relation R on A is asymmetric iff ∀ a,b ∈ A
if (a,b) ∈ R then (b,a) ∉ R.
A relation R on A is transitive iff ∀ a,b,c ∈ A
if (a,b) ∈ R and (b,c) ∈ R then (a,c) ∈ R.
Examples
Let A = {1, 2, 3, 4} and define the relation "|" (= R) on A to be
integer division.
| = {(1,1), (1,2), (1,3), (1,4), (2,2), (2,4), (3,3), (4,4)}.
This relation is reflexive, antisymmetric and transitive.
Let A = {1,2,3,4} and define the relation ">" on A by:
> = {(2,1), (3,2), (3,1), (4,3), (4,2), (4,1)}
This relation is asymmetric, antisymmetric and transitive.
Let A = {1, 2, 3, 4} and R = {(1,1), (2,3), (3,2), (4,2)}.
This relation has none of the listed properties.
Examples
Let A = {1, 2, 3} and R = {(1,1), (2,2), (3,3), (1,2), (2,1)}
This relation is reflexive, symmetric and transitive.
Let A = {1, 2, 3} and R = {(1,1), (2,2)}
This relation is symmetric, antisymmetric and transitive.
Properties and Digraphs
Relation Property
In Digraph you see:
Reflexive
There is a self-loop at every vertex
Symmetric
Whenever there is an arc from a to b, there
is also one from b to a.
Asymmetric
Whenever there is an arc from a to b, there
is never one from b to a & no self-loops.
Antisymmetric
Whenever there is an arc from a to b, there
is never one from b to a, if a ≠ b.
Transitive
Whenever there is an arc from a to b and
one from b to c, there is one from a to c.
Properties and Digraphs
Let A = {1, 2, 3, 4} and define the relation "|" (= R) on A to be
integer division.
| = {(1,1), (1,2), (1,3), (1,4), (2,2), (2,4), (3,3), (4,4)}.
This relation is reflexive, antisymmetric and transitive.
1
4
2
3