RELATIONSPearl Rose Cajenta

REPORTER

What is a 'relation'?In math, a relation is just a set of

ordered pairs.

- is a pair of numbers used tolocate a point on a coordinate plane;the first number tells how far to movehorizontally and the second numbertells how far to move vertically.

*Ordered Pair

*Set- is a collection.

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) }

Given a sets A and B, a binary relation from A to

B is a set of ordered pairs (a,b), whose entries a ϵA and b ϵ B. Each ordered pair (a,b) in arelation is a member of the Cartesian set A xB. Hence,a relation from A to B is asubset of A x B.

Click for some Examples

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

Example 1:of the Domain and Range

RELATION {(0,1) , (3,22) , (90, 34)}Domain : 0 3 90 Range: 1 22 34

Example 2: Domain and range of a relation

RELATION {(2,-5) , (4,31) , (11, -11), (-

21,3)}

Domain : 2 4 11 -21

Range: -5 31 -11 3

Example 3What is the domain and range of the following relation?

ANSWER:

Arrow Diagram

-is use by describing the relation from A into B sets.

Example:

Let A = { 1, 2, 3} and B = { 6, 7, 8, 9}. We defined the relation R by the set of ordered pairs

R= {( 1, 6),( 1, 7),( 2, 6),( 2, 8),( 2, 9),( 3, 9)}

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2

3

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Directed Graph

Is a set of vertices and a collectionof directed edges that eachconnects an ordered pair ofvertices.

Let A = {1,2,3}. Let R be the relation defined by the following sets.

R= { (1,1), (1,2), (2,1), (3,1), (3,3)}Let us construct the digraph of the relation.

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1 2

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Examples

1.We can take the set A of students and the setB of programs. We Can relate the elements of Ato the elements of B by defining that an element aof 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,InformationTechnology),(Recca,InformationScience),(Jean,CivilEngeneering)

More Examples

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} = 0R6={(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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2. Let A = {1,2,3,4} and B = {u,v,w}, LetR = {(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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Let A={ 1, 3 } and B= { 2, 5 }. Then weask how elements in A are related toelements in B via the inequality `` ''.The answer is

1 2,1 5, 3 2, 3 5 .

R= { (1, 2), (1, 5), (3 ,5) } ,

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Let A ={ 1,2} and B={ 1,2,3} and define a binaryrelation 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-yis even. This is done in the following table

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(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.

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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. >>>>>