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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

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) | ab}
- 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+b2}

- 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

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