Processing math: 100%
Skip to main content
Library homepage
 

Text Color

Text Size

 

Margin Size

 

Font Type

Enable Dyslexic Font
Mathematics LibreTexts

6.2: Properties of Relations

( \newcommand{\kernel}{\mathrm{null}\,}\)

Note: If we say R is a relation "on set A" this means R is a relation from A to A; in other words, RA×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 aA, aRa.

example: consider D:ZZ by xDyx|y. Since a|a for all aZ the relation D is reflexive.

Definition: Symmetric Property

A relation R on A is  symmetric if and only if for all a,bA, if aRb, then bRa.

Clearly the relation = is symmetric since x=yy=x.  However, divides is not symmetric, since 510 but 105.

Definition: Transitive Property

A relation R on A is  transitive if and only if for all a,b,cA, if aRb and bRc, then aRc.

example: consider G:RR by xGyx>y.  Since if a>b and b>c then a>c is true for all a,b,cR, 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 aRbab. Determine whether R is reflexive, symmetric, or transitive.

hands-on exercise 6.2.2

The relation S on the set R is defined as aSbab>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 aTbabQ.

Since aa=1Q, 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,bcQ, then ab=mn and bc=pq for some nonzero integers m, n, p, and q. Then ac=abbc=mpnqQ. 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 aUb5(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)VST.

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: aWba and b have the same last name.

Example 6.2.8 Congruence Modulo 5

Consider the relation R on Z defined by xRy5(xy).

Note:  (1) R is called Congruence Modulo 5.  (2) We have proved  amod5=bmod55(ab).

Prove R is an equivalence relation.

Proof:

Reflexive: Consider any integer aaa=050 by the definition of divides since 5(0)=0 and 0Z.
So, 5(a=a) thus aRa by definition of R.
R is reflexive.

Symmetric: Let a,bZ such that aRb.  We must show that bRa.
Since aRb,  5(ab) by definition of R. By definition of divides, there exists an integer k such that 5k=ab. 
By algebra: 5k=ba 5(k)=ba.
kZ since the set of integers is closed under multiplication. So, 5(ba) by definition of divides. bRa by definition of R.
R is symmetric.

Transitive: Let a,b,cZ such that aRb and bRc.  We must show that aRc.
 5(ab) and 5(bc) by definition of R. By definition of divides, there exists an integers j,k such that 5j=ab.5k=bc. Adding the equations together and using algebra: 5j+5k=ac5(j+k)=ac.  j+kZ since the set of integers is closed under addition. So, 5(ac) 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 xA.
  • 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,yA.
  • 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)AXY=. 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}S1S2= and  S2S3=, but S1S3. 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)xy 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)xy is even }

c) V3={(x,y)x is a multiple of y}


This page titled 6.2: Properties of Relations is shared under a CC BY-NC-SA license and was authored, remixed, and/or curated by Harris Kwong (OpenSUNY) .

Support Center

How can we help?