6.2: Properties of Relations
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Note: If we say R is a relation "on set A" this means R is a relation from A to A; in other words, R⊆A×A.
We will define three properties which a relation might have.
Definition: Reflexive Property
A relation R on A is reflexive if and only if for all a∈A, aRa.
example: consider D:Z→Z by xDy⟺x|y. Since a|a for all a∈Z the relation D is reflexive.
Definition: Symmetric Property
A relation R on A is symmetric if and only if for all a,b∈A, if aRb, then bRa.
Clearly the relation = is symmetric since x=y→y=x. However, divides is not symmetric, since 5∣10 but 10∤5.
Definition: Transitive Property
A relation R on A is transitive if and only if for all a,b,c∈A, if aRb and bRc, then aRc.
example: consider G:R→R by xGy⟺x>y. Since if a>b and b>c then a>c is true for all a,b,c∈R, the relation G is transitive.
Example 6.2.1
B is a relation on all people on Earth defined by xBy if and only if x is a brother of y.
- Reflexive?
-
No, Jamal is not a brother to himself.
- Symmetric?
-
No, Jamal can be the brother of Elaine, but Elaine is not the brother of Jamal.
- Transitive?
-
Yes, if X is the brother of Y and Y is the brother of Z , then X is the brother of Z.
Example 6.2.2
Consider the relation R on the set A={1,2,3,4} defined by R={(1,1),(2,3),(2,4),(3,3),(3,4)}.
- Reflexive?
-
No, since (2,2)∉R, the relation is not reflexive.
- Symmetric?
-
No, we have (2,3)∈R but (3,2)∉R, thus R is not symmetric.
- Transitive?
-
Yes. By going through all the ordered pairs in R, we verify that whether (a,b)∈R and (b,c)∈R, we always have (a,c)∈R as well. This shows that R is transitive.
Example 6.2.3
Define the relation S on the set A={1,2,3,4} according to S={(2,3),(3,2)}.
- Reflexive?
-
No, since (2,2)∉R, the relation is not reflexive.
- Symmetric?
-
Yes. Since we have only two ordered pairs, and it is clear that whenever (a,b)∈S, we also have (b,a)∈S. Hence, S is symmetric.
- Transitive?
-
Since (2,3)∈S and (3,2)∈S, but (2,2)∉S, the relation S is not transitive.
hands-on exercise 6.2.1
Define the relation R on the set R as aRb⇔a≤b. Determine whether R is reflexive, symmetric, or transitive.
hands-on exercise 6.2.2
The relation S on the set R∗ is defined as aSb⇔ab>0. Determine whether S is reflexive, symmetric, or transitive.
Example 6.2.4
Here are two examples from geometry. Let T be the set of triangles that can be drawn on a plane. Define a relation S on T such that (T1,T2)∈S if and only if the two triangles are similar. It is easy to check that S is reflexive, symmetric, and transitive.
Let L be the set of all the (straight) lines on a plane. Define a relation P on L according to (L1,L2)∈P if and only if L1 and L2 are parallel lines. Again, it is obvious that P is reflexive, symmetric, and transitive.
Example 6.2.5
The relation T on R∗ is defined as aTb⇔ab∈Q.
Since aa=1∈Q, the relation T is reflexive.
The relation T is symmetric, because if ab can be written as mn for some nonzero integers m and n, then so is its reciprocal ba, because ba=nm.
If ab,bc∈Q, then ab=mn and bc=pq for some nonzero integers m, n, p, and q. Then ac=ab⋅bc=mpnq∈Q. Hence, T is transitive.
Therefore, the relation T is reflexive, symmetric, and transitive.
Definition: Equivalence Relation
A relation is an equivalence relation if and only if the relation is reflexive, symmetric and transitive.
Example 6.2.6
The relation U on Z is defined as aUb⇔5∣(a+b).
Is U an equivalence relation?
- Answer
-
If 5∣(a+b), it is obvious that 5∣(b+a) because a+b=b+a. Thus, U is symmetric.
However, U is not reflexive, because 5∤(1+1).
Therefore U is not an equivalence relation
hands-on exercise 6.2.3
Determine whether the following relation V on some universal set U is an equivalence relation: (S,T)∈V⇔S⊆T.
Example 6.2.7
Consider the relation V on the set A={0,1} is defined according to V={(0,0),(1,1)}.
Is V an equivalence relation?
- Answer
-
The relation V is reflexive, because (0,0)∈V and (1,1)∈V.
It is clearly symmetric, because (a,b)∈V always implies (b,a)∈V.
Because V consists of only two ordered pairs, both of them in the form of (a,a), V is transitive.
Therefore, V is an equivalence relation.
hands-on exercise 6.2.4
Determine whether the following relation W on a nonempty set of individuals in a community is an equivalence relation: aWb⇔a and b have the same last name.
Example 6.2.8 Congruence Modulo 5
Consider the relation R on Z defined by xRy⟺5∣(x−y).
Note: (1) R is called Congruence Modulo 5. (2) We have proved amod5=bmod5⟺5∣(a−b).
Prove R is an equivalence relation.
- Proof:
-
Reflexive: Consider any integer a. a−a=0. 5∣0 by the definition of divides since 5(0)=0 and 0∈Z.
So, 5∣(a=a) thus aRa by definition of R.
∴R is reflexive.
Symmetric: Let a,b∈Z such that aRb. We must show that bRa.
Since aRb, 5∣(a−b) by definition of R. By definition of divides, there exists an integer k such that 5k=a−b.
By algebra: −5k=b−a 5(−k)=b−a.
−k∈Z since the set of integers is closed under multiplication. So, 5∣(b−a) by definition of divides. bRa by definition of R.
∴R is symmetric.
Transitive: Let a,b,c∈Z such that aRb and bRc. We must show that aRc.
5∣(a−b) and 5∣(b−c) by definition of R. By definition of divides, there exists an integers j,k such that 5j=a−b.5k=b−c. Adding the equations together and using algebra: 5j+5k=a−c5(j+k)=a−c. j+k∈Z since the set of integers is closed under addition. So, 5∣(a−c) by definition of divides. aRc by definition of R.
∴R is transitive.
Thus, by definition of equivalence relation, R is an equivalence relation.
Summary and Review
- A relation from a set A to itself is called a relation on set A.
- Given any relation R on a set A, we are interested in three properties that R may or may not have.
- The relation R is said to be reflexive if every element is related to itself, that is, if xRx for every x∈A.
- The relation R is said to be symmetric if the relation can go in both directions, that is, if xRy implies yRx for any x,y∈A.
- Finally, a relation is said to be transitive if we can pass along the relation and relate two elements if they are related via a third element.
- More precisely, R is transitive if xRy and yRz implies that xRz.
Exercises
Exercise 6.2.1
Let S be a nonempty set and define the relation A on P(S) by (X,Y)∈A⇔X∩Y=∅. It is clear that A is symmetric.
a) Explain why A is not reflexive.
b) Is A transitive? Explain.
c) Let S={a,b,c}. Draw the directed (arrow) graph for A.
- Answer:
-
(a) Since set S is not empty, there exists at least one element in S, call one of the elements x. The power set must include {x} and {x}∩{x}={x} and thus is not empty. So we have shown an element which is not related to itself; thus S is not reflexive.
(b) Consider these possible elements of the power set: S1={w,x,y},S2={a,b},S3={w,x}. S1∩S2=∅ and S2∩S3=∅, but S1∩S3≠∅. We have shown a counter example to transitivity, so A is not transitive.
(c) Here's a sketch of some of the diagram should look:
-There are eight elements on the left and eight elements on the right
-This relation is symmetric, so every arrow has a matching cousin. i.e there is \{a,c\}\right arrow\{b}\} and also \{b\}\right arrow\{a,c}\}.
-The empty set is related to all elements including itself; every element is related to the empty set.
Exercise 6.2.2
For each of these relations on N−{1}, determine which of the three properties are satisfied.
a) A1={(x,y)∣x and y are relatively prime}
b) A2={(x,y)∣x and y are not relatively prime}Exercise 6.2.3
For each of the following relations on N, determine which of the three properties are satisfied.
a) B1={(x,y)∣x divides y}
b) B2={(x,y)∣x+y is even }
c) B3={(x,y)∣xy is even }
- Answer:
-
(a) reflexive, transitive
(b) reflexive, symmetric, transitive
(c) symmetric
Exercise 6.2.4
For each of the following relations on N, determine which of the three properties are satisfied.
a) D1={(x,y)∣x+y is odd }
b) D2={(x,y)∣xy is odd }
Exercise 6.2.5
For each of the following relations on Z, determine which of the three properties are satisfied.
a) U1={(x,y)∣3 divides x+2y}
b) U2={(x,y)∣x−y is odd }
- Answer:
-
(a) reflexive, symmetric and transitive (try proving this!)
(b) symmetric
Exercise 6.2.6
For each of the following relations on Z, determine which of the three properties are satisfied.
a) V1={(x,y)∣xy>0}
b) V2={(x,y)∣x−y is even }
c) V3={(x,y)∣x is a multiple of y}