Section 7.1

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Section 7.1. Relations and their properties. Binary relation. A binary relation is a set of ordered pairs that expresses a relationship between elements of 2 sets Formal definition: Let A and B be sets A binary relation from A to B is a subset of AxB (Cartesian product).

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### Section 7.1

Relations and their properties

Binary relation
• A binary relation is a set of ordered pairs that expresses a relationship between elements of 2 sets
• Formal definition:
• Let A and B be sets
• A binary relation from A to B is a subset of AxB (Cartesian product)
Denotation of binary relation R
• Suppose a  A and b  B
• If (a,b)  R, then aRb
• If (a,b)  R, then aRb
• If aRb, we can state that a is related to b by R
Example 1
• Let A = {0,1,2} and B = {a,b}
• Then {(0,a), (0,b), (1,a), (2,b)} is a relation from A to B
• We can state that, for instance, 0Ra and 1Rb
• We can represent relations graphically, as shown on the next slide
Example 1

A = {0,1,2}

B = {a,b}

R = {(0,a), (0,b), (1,a), (2,b)}

R a b

0 x x

1 x

2 x

0

a

1

b

2

Functions as relations
• A function f from set A to set B assigns a unique element of B to each element of A
• The graph of f is the set of ordered pairs (a,b) such that b = f(a)
• The graph of f is a subset of AxB, so it is a relation from A to B
Functions as relations
• The graph of f has the property that every element of A is the first element of exactly one ordered pair of the graph
• If R is a relation from A to B such that every element is the first element of exactly one ordered pair of R, then a function can be defined with R as its graph
Not all relations are functions
• A relation can express a one-to-many relationship between elements of sets A and B, where an element of A may be related to several elements of B
• On the other hand, a function represents a relation in which exactly one element of B is related to each element of A
Relations on a set
• A relation on a set A is a relation from A to A; in other words, a subset of AxA
• Example: Let A = {1,2,3,4,5,6}; which ordered pairs are in the relation R={(a,b)|a divides b}?
• Solution:

{(1,1) (1,2), (1,3), (1,4), (1,5), (1,6), (2,2), (2,4), (2,6), (3,3), (3,6), (4,4), (5,5), 6,6)}

Relations on the set of integers
• Relations on the set of integers are infinite relations
• Some examples include:

R1 = {(a,b) | a = b}

R2 = {(a,b) | a = 5b}

R3 = {(a,b) | a = b+2}

Finding the number of relations on a finite set
• A relation on a set A is a subset of AxA
• If A has n elements, AxA has n2 elements
• A set with m elements has 2m subsets
• Therefore, there are 2n2 relations on a set with n elements
• For set {a,b,c,d} there are 216, or 65,536 relations on the set
Properties of relations
• Reflexive: a relation R on set A is reflexive if (a,a)  R for every element a  A
• For example, for set A = {1,2,3}
• if R = {(1,1), (1,2), (2,2), (3,1), (3,3)} then R is a reflexive relation
• On the other hand, if R = {(1,1), (1,2), (2,3), (3,3)} then R is not a reflexive relation
Properties of relations
• Symmetric: a relation R on a set A is symmetric if (b,a)  R whenever (a,b)  R for all a,b  A
• For set A = {a,b,c,d}:
• if R = {(a,b), (b,a), (c,d), (d,c)} then R is symmetric
• if R = {(a,b), (b,a), (c,d), (c,b)} then R is not symmetric
Properties of relations
• Antisymmetric: a relation R on a set A is antisymmetric if (a,b)  R and (b,a)  R only when a=b
• Note that symmetric and antisymmetric are not necessarily opposite; a relation can be both at the same time
Examples of symmetry and antisymmetry
• For A={1,2,3}:
• R = {(1,1), (1,2), (2,1)} is symmetric but not antisymmetric
• R = {(1,1), (1,2), (2,3)} is antisymmetric but not symmetric
• R = {} is both symmetric and antisymmetric
• R = {(1,2), (1,3), (2,3)} is antisymmetric
Properties of relations
• Transitive: A relation R on a set A is called transitive if, whenever (a,b)  R and (b,c)  R, then (a,c)  R for a,b,c  A
• For set A = {1, 2, 3, 4}:
• R = {(1,3), (3,4), (1,2), (2,3), (2,4), (1,4)} is transitive
• R = {(1,3), (3,4), (1,2), (2,4)} is not transitive
Example 2
• Let A = set of integers and
• R1 = {(a,b) | ab}
• R2 = {(a,b) | a<b}
• R3 = {(a,b) | a=b or a=-b}
• R4 = {(a,b) | a=b}
• R5 = {(a,b) | a=b+1}
• R6 = {(a,b) | a+b2}
• Which of these are reflexive, symmetric, antisymmetric, transitive?
Combining relations
• Since relations from A to B are subsets of AxB, relations from A to B can be combined any way 2 sets can be combined
• Let A={1,2,3} and B={1,2,3,4} and R1={(1,1), (2,2), (3,3)}, R2={(1,1),(1,2),(1,3),(1,4)}
• R1  R2 = {(1,1), (1,2), (2,2), (1,3),(3,3), (1,4)}
• R1  R2 = {(1,1)}
• R1 - R2 = {(2,2), (3,3)}
• R2 - R1 = {(1,2), (1,3), (1,4)}
Composition of relations
• Let R be a relation from A to B and S be a relation from B to C
• S  R is the relation consisting of ordered pairs (a,c) where a  A and c  C and there exists an element b  B such that (a,b)  R and (b,c)  S