Friday, March 27
Today we will continue in Course Notes Chapter 7.
Last time, we began to define properties of relations on a set A.
In particular, we defined the properties reflexive, irreflexive, and symmetric.
Which of these relations on the positive integers is both reflexive and symmetric?
A. R = {(x, y): x º y (mod 5)}
B. S = {(x, y): x | y}
C. Both R and S.
D. None of these
Special properties of relations R: A®A
Last week we defined these properties:
Reflexive ∀x⎡ x,x ∈ R⎤ ⎣( ) ⎦ Not reflexive ∃x ⎡ x,x ∉R⎤ ⎣( ) ⎦
Irreflexive ∀x⎡ x,x ∉ R⎤ ⎣( ) ⎦ Not irreflexive ∃x ⎡ x,x ∈R⎤ ⎣( ) ⎦
Symmetric ∀x∀y⎡ x, y ∈ R → y,x ∈ R⎤ ⎣( ) ( ) ⎦
Not symmetric ∃x∃y ⎡ x, y ∈R ∧ y,x ∉R⎤ ⎣( ) ( ) ⎦
Suppose a finite relation is symmetric.
What do we know about the digraph?
What do we know about the connection matrix?
Asymmetric, Antisymmetric relations
The “<” relation on the real numbers has another property that is of general interest:
For all real numbers x, y, if x is related to y, then y isn’t related to x.
A relation with a property like that is called asymmetric.
Formally:
A relation R on a set A is asymmetric if and only if ______
Contrast the “<” relation on the real numbers with the “≤” relation.
Suppose x, y are real numbers such that x ≤ y and y ≤ x. Then what do we know about x and y?
Definition: A relation R on a set A is antisymmetric if and only if
______
Let R be a relation on a set A. Select the true statement(s), if any:
Statement 1 Statement 2 A. Only Statement 1 If R is asymmetric, If R is antisymmetric, is true. then R is then R is asymmetric. B. Only Statement 2 antisymmetric. is true. C. Both are true. D. Both are false.
How does asymmetric differ from antisymmetric?
What these two properties have in common, is that in either case, relationships can only go in “one direction.”
That is (for a relation R that is asymmetric or antisymmetric), if a and b are distinct elements and (a, b) Î R, then (b, a) Ï R.
The difference between the two properties is that in an antisymmetric relation, if is ok (but not necessary) for an element to be related to itself. In an asymmetric relation, an element cannot be related to itself.
In terms of finite relations and digraphs, for both of these properties, if there is an arrow from vertex a to vertex b, there cannot be a return arrow from b to a.
However, an antisymmetric relation allows loops, but an asymmetric relation does not allow loops.
In terms of familiar relations, the < relation on real numbers is asymmetric (and antisymmetric), but the ≤ relation is antisymmetric, but not asymmetric.
Transitive relations
Consider the “|” relation in arithmetic again (let the domain and codomain be the set of positive integers).
A typical exercise from Test 2 topics was to prove, for all integers a, b, c:
If a|b and b|c then a|c.
Note, for example, that 2|4 and 4|24, and that 2|24.
More generally, "x"y"z [(xRy Ù yRz) ® xRz]
Any relation that has that property is called transitive.
Here is an example of a relation involving numbers that is not transitive:
For real numbers a, b, let R be the relation “a is the square root of b.”
That is, R = {(a,b) :a = b}
Note that 2R4 and 4R16, but 2R16.
When discussing methods of proof, we proved that congruence mod m is a transitive relation on !.
Let R be the following relation on the real numbers: xRy iff x – y is rational.
Consider these possible properties: reflexive, irreflexive, symmetric, asymmetric, antisymmetric, transitive.
How many of these properties does R possess?
A. 0 B. 1 C. 2 D. 3 E. 4 or more
Now, prove your answer, using the definition of rational number.
Let A = {1, 2, 3, 4} and let R be this relation on A: R = {(1, 2), (2, 2), (4, 3), (4, 4)}.
Consider these possible properties: reflexive, irreflexive, symmetric, asymmetric, antisymmetric, transitive.
How many of these properties does R possess?
A. 0 B. 1 C. 2 D. 3 E. 4 or more
Relations and set mathematics
Because relations are sets (sets of ordered pairs), all of the familiar concepts from set mathematics apply to the study of relations.
For instance, relations can be combined using set operations such as union, intersection, difference and so on.
Likewise, one relation might be a subset of another relation.
EXAMPLE
Let R, S, be these relations on !:
R = x, y : x ≡ y mod10 S = x, y : x2 ≡ y2 mod10 {( ) ( )} {( ) ( )}
Prove/disprove
R ⊆ S