Discrete Math

1. RELATIONS
PearlRoseCajenta
REPORTER

2. What is a 'relation'?
In math, a relation is just a set of
ordered pairs.
- is a pair of numbers used to
locate a point on a coordinate plane;
the first number tells how far to move
horizontally and the second number
tells how far to move vertically.
*OrderedPair
*Set
- is a collection.

3. Note:
{ } are the symbol for "set“
Some Examples of Relations include:
{ (0,1) , (55,22), (3,-50) }
{ (0, 1) , (5, 2), (-3, 9) }
{ (-1,7) , (1, 7), (33, 7), (32, 7) }

4. Given a sets A and B, a binary relation from A to
B is a set of ordered pairs (a,b), whose entries a ϵ
A and b ϵ B. Each orderedpair (a,b) in a
relation is a memberof the Cartesian set A x
B. Hence,arelation from A to B is a
subset of A x B.
Clickfor some Examples

5. Domain and Range of a
‘Relation’
The Domain:
Is the set of all the first numbers of the ordered
pairs.In other words, the domain is all of the x-
values.
The Range:
Is the set of the second numbers in each pair,
or the y-values

6. Example 1:
of the Domain and Range
RELATION {(0,1) , (3,22) , (90, 34)}
Domain : 0 3 90
Range: 1 22 34

7. Example 2:
Domain and rangeof a relation
RELATION {(2,-5) , (4,31) , (11, -11), (-
21,3)}
Domain: 2 4 11 -21
Range: -5 31 -11 3

8. Example3
What is the domain and range of the followingrelation?
ANSWER:

9. Arrow Diagram
-is use by describingthe
relationfrom A into B sets.

10. Example:
Let A = { 1, 2, 3} and B = { 6, 7, 8, 9}. We defined the
relationR by the set of orderedpairs
R= {( 1, 6),( 1, 7),( 2, 6),( 2, 8),( 2, 9),( 3, 9)}
1
2
3
6
7
8
9

11. DirectedGraph
Is a set of vertices and a collection
of directed edges that each
connects an ordered pair of
vertices.

12. Let A = {1,2,3}. Let R be the relationdefined by the
followingsets.
R= { (1,1), (1,2), (2,1),(3,1), (3,3)}
Let us construct the digraphof the relation.
3
1 2
NEXT

14. Examples
1.We can take the set A of students and the set
B of programs. We Can relate the elements of A
to the elements of B by defining that an element a
of A is related to an element b of B if a takes b.
A= { Beth, Mita, Recca, Jean}
B={Computer Science, Information Technology,
Information Science, Civil Engineering}
Answer:
R={(Beth,ComputerScience),(Mita,InformationTechn
ology),(Recca,InformationScience),(Jean,Civil
Engeneering)
More Examples

15. Let A = {1,2,3,}, then each of the following is a
relation in A:
R1 ={(a,b)| a < b} = {(1,2),(1,3),(2,3)}
R2={(a,b)| a = b} = {(1,1),(2,2),(3,3)}
R3={(a,b)| a > b} = {(2,1),(3,1),(3,2)}
R4 ={(a,b)| a + b =4} = {(1,3),(2,2),(3,1)}
R5={(a,b)| b = 4a} = 0
R6={(1,1),(1,2),(1,3),(2,1),(2,2),(2,3),(3,1),
(3,2),(3,3)}
R2 is called the identity relation. R5 is called trivial
relation or void relation or empty relation. The
universal relation on a set consist of all possible
ordered pairs of elements of a set, example is R6.
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17. 2. Let A = {1,2,3,4} and B = {u,v,w}, Let
R = {(1,u),(2,u),(3,v),(4,w)}
Then R is a subset of A x B so R is a relation from
A to B. Notice 1Ru but 3R u.
3.Let A = {1,2,3,4} and B = {1,2 ,3,4}, where R is a
relation from A to B defined by a|b, “a divides b,
(a,b) ϵ R. We see that
R = {(1,1),(1,2),(1,3),(1,4),(2,2),(2,4),(3,3),(4,4)}
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18. Let A={ 1, 3 } and B= { 2, 5 }. Then we
ask how elements in A are related to
elements in B via the inequality `` ''.
The answer is
1 2,1 5, 3 2, 3 5 .
R= { (1, 2), (1, 5), (3 ,5) } ,
NEXT

19. Let A ={ 1,2} and B={ 1,2,3} and define a binary
relation R from A to B by for any (x,y) AxB: (x,y)
R iff x-y is even
(a) Give R by its explicit elements.
(b) Is 1R1 ? Is 2R3 ? Is 1R3 ?
Solution
(a) For any (x,y) pair in AXB ={ (1,1), (1,2), (1,3),
(2,1), (2,2), (2,3) }, we must check if xRy or if x-y
is even. This is done in the following table
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20. (x,y) property of x-y conclusion
(1,1) 1-1 even (1,1) R
(1,2) 1-2 odd (1,2) R
(1,3) 1-3 even (1,3) R
(2,1) 2-1 odd (2,1) R
(2,2) 2-2 oven (2,2) R
(2,3) 2-3 odd (2,3) R
Hence R={ (1,1), (1,3), (2,2)}.
(b)
Yes. 1R1 since (1,1) R.
No. 2R3 since (2,3) R.
Yes. 1R3 since (1,3) R.
NEXT

21. Let A={1,2,3} and a binary relation E be defined by
(x,y) E iffx-y is even and x,y A. Then E={ (1,1),
(2,2), (3,3), (1,3), (3,1)} can be represented by the
following digraph
(2,2):
2-2=0 even, hence the loop at vertex labelled by 2
A.
(1,3):
1-3=-2 even, hence the arrow from vertex 1 to
vertex 3. >>>>>