CmSc175 Discrete Mathematics
Lesson 16: Basic Properties of Binary Relations on a Set
Definitions:
Let R be a binary relation on a set A.
1. R is reflexive, iff for all x  A, (x,x)  R, i.e. xRx is true.
2. R is symmetric, iff for all x, y  A, if (x, y)  R, then (y, x)  R
i.e xRy  yRx is true
3. R is transitive iff for all x, y, z  A, if (x, y)  R and (y,z)  R , then (x, z)  R
i.e. (xRy  yRz)  xRz is true
A. Reflexive relations
Let R be a binary relation on a set A.
R is reflexive, iff for all x  A, (x,x)  R, i.e. xRx is true.
1. Examples:
1. Equality is a reflexive relation
for any object x:
x = x is true.
2. "less then" (defined on the set of real numbers) is not a reflexive relation.
for any number x:
x < x is not true
3. " less then or equal to" (defined on the set of real numbers) is a reflexive relation
for any number x
x  x is true
4. A = {1,2,3,4}, R = {(1,1), (1,2), (2,2), (2,3), (3,3), (3,4), (4,4)}
5. A = {1,2,3,4}, R = {(1,2), (2,3), (3,4), (4,1)} - not reflexive
6. A = {1,2,3,4}, R = {(1,1), (1,2), (3,4), (4,4)} - not reflexive
2. Graph representation of reflexive relations
Rule: if xRx is true, there is a loop on node x.
Example:
A:= {1,2,3}
R = "less then or equal to"
R = {(1,1),(1,2),(1,3),(2,2),(2,3),(3,3)}
1
1
2
3
3. Matrix representation of reflexive relations
1
2
3
1
2
3
1
0
0
1
1
0
1
1
1
4. Reflexive and irreflexive relations
Compare the three examples below:
1. A = {1,2,3,4}, R1 = {(1,1), (1,2), (2,2), (2,3), (3,3), (3,4), (4,4)}
2. A = {1,2,3,4}, R2 = {(1,2), (2,3), (3,4), (4,1)}
3. A = {1,2,3,4}, R3= {(1,1), (1,2), (3,4), (4,4)}
R1 is a reflexive relation. R2 ? R3 ?
Definition: Let R be a binary relation on a set A.
R is irreflexive iff for all x  A, (x,x)  R
Definition: Let R be a binary relation on a set A.
R is neither reflexive, nor irreflexive iff
there is x  A, such that (x, x)  R. and there is y  A such that (y, y)  R
Thus R2 is irreflexive, R3 is neither reflexive nor irreflexive.
reflexive:
irreflexive:
neither:
for all x: xRx
for no x: xRx
for some x: xRx is true, for some y: yRy is false
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B. Symmetric relations
R is symmetric, iff for all x, y  A, if (x, y)  R, then (y, x)  R
i.e xRy  yRx is true
This means: if two elements x and y are in relation R, then y and x are also in R, i.e. if
xRy is true, yRx is also true.
1. Examples:
1.
2.
3.
4.
5.
6.
7.
8.
equality is a symmetric relation: if a = b then b = a
"less than" is not a symmetric relation : if a < b is true then b < a is false
"sister" on the set of females is symmetric
"sister" on the set of all human beings is not symmetric
"friends" - symmetric: friend(a,b)  friend(b,a)
A = {1,2,3,4}, R1 = {(1,1), (1,2), (2,1), (2,3), (3,2), (4,4)}
A = {1,2,3,4}, R2 = {(1,1), (1,2), (2,1) , (2,3), (4,4)} - not symmetric
A = {1,2,3,4}, R3 = {(1,1), (1,2), (2,3), (4,4)} - not symmetric
2. Graph representation of symmetric relations
Rule: if R is a symmetric relation, all links are bi-directional
Example:
friend(x,y), x,y  {Ann, Tim, Paul, Jane, Jim}
Ann
Tim
Paul
Jane
Jim
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3. Matrix representation of symmetric relations
Ann
Tim
Paul Jane Jim
Ann
0
1
0
0
0
Tim
1
0
1
0
0
Paul 0
1
0
0
0
Jane 0
0
0
0
1
Jim
0
0
1
0
0
The matrix is symmetric.
4. Symmetric and anti-symmetric relations
Compare the relations:
1. A = {1,2,3,4}, R1 = {(1,1), (1,2), (2,1), (2,3), (3,2), (4,4)}
2. A = {1,2,3,4}, R2 = {(1,1), (1,2), (2,3), (4,4)}
3. A = {1,2,3,4}, R3 = {(1,1), (1,2), (2,1) , (2,3), (4,4)}
Definition: Let R be a binary relation on a set A.
R is anti-symmetric if for all x, y  A, x  y, if (x, y)  R, then (y, x)  R.
Definition: R is neither symmetric nor anti-symmetric iff it is not symmetric and not
anti-symmetric.
symmetric:
xRy  yRx for all x and y
anti-symmetric:
xRy and yRx  x = y
neither: for some x and y: xRy, and yRx
for others xRy is true, yRx is not true
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C. Transitive relations
Let R be a binary relation on a set A.
R is transitive iff for all x, y, z  A, if (x, y)  R and (y,z)  R , then (x, z)  R
i.e. (xRy  yRz)  xRz is true
1. Examples:
1. Equality is a transitive relation a = b, b = c, hence a = c
2. "less than" is a transitive relation a < b, b < c, hence a < c
3. mother_of(x,y) is not a transitive relation
4. sister(x,y) is a transitive relation
5. brother (x,y) is a transitive relation.
6. A = {1,2,3,4} R = {(1,1), (1,2), (1,3), (2,3), (4,3)} - transitive
7. A = {1,2,3,4} R = {(1,1), (1,2), (1,3), (2,3), (3,4)} - not transitive
2. Graph representation of transitive relations
Rule: if there is a link from a to b, and a link from b to c, then there must be a link from a
to c.
Example:
A = {1,2,3,4}
R = {(1,2), (1,3), (1,4),(2,3),(2,4),(3,4)}
This is the relation "less than"
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1
3
4
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Problems:
1. Determine the relevant properties of the following relations on the set A = {1,2,3}
R1 = {(1,1), (1,2), (2,1), (2,2), (2,3), (3,2), (3,3)}
Reflexive
irreflexive
neither
Symmetric
anti-symmetric
neither
Transitive
not transitive
R2 = {(1,2), (2,3), (3,2)}
Reflexive
irreflexive
neither
Symmetric
anti-symmetric
neither
Transitive
not transitive
R3 = {(1,2),(2,2), (2,3), (1,3)}
Reflexive
irreflexive
neither
Symmetric
anti-symmetric
neither
Transitive
not transitive
2. Give examples of relations on the set A = {1,2,3,4} with the following properties:
a. reflexive, symmetric, not transitive
b. reflexive, neither symmetric nor anti-symmetric, not transitive
c. reflexive, anti-symmetric, transitive
d. neither reflexive not irreflexive, symmetric, transitive.
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3. Determine the properties (reflexivity, symmetry, transitivity) of the following relations
defined on the set of natural numbers:
a. R = {(x, y) | x = y2 }
b. R = {(x, y) | x > y }
c. R = {(x, y) | x  y }
d. R = {(x, y) |  p : x – y = 3p }
e. R = {(x, y) |  p : x + 2y = 3p }
4. Let X be a nonempty set. Define a relation on P(X), the power set of X, as
R = {(A, B) | A  B}
Determine the properties of R.
5. Let R and S be reflexive relations on a set X. Prove or disprove the reflexivity of the
following relations:
a. R  S
b. R  S
c. R – S
d. R -1
6. Let R and S be symmetric relations on a set X. Prove or disprove the symmetry of the
following relations:
a. R  S
b. R  S
c. R – S
d. R -1
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7. Let R and S be anti-symmetric relations on a set X. Prove or disprove the antisymmetry of the following relations:
a. R  S
b. R  S
c. R – S
d. R -1
8. Let R and S be transitive relations on a set X. Prove or disprove the transitivity of the
following relations:
a. R  S
b. R  S
c. R – S
d. R -1
9. Let R and S be non-transitive relations on a set X. Prove or disprove the nontransitivity of the following relations:
a. R  S
b. R  S
c. R – S
d. R -1
10. What is wrong with the following proof, which supposedly shows that any relation R
on X that is symmetric and transitive is reflexive?
Let x, y  X. Since R is symmetric, we have: (x, y)  R and (y, x)  R. Since R
is transitive, we have (x, x)  R. Therefore R is reflexive.
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