Relation and function

Relation

1. Relation from set A to set B is the relationship which connects the elements of set A to the elements of set B

2. The relation between two sets can be expressed by:

1) Arrow Diagram 2) Cartesian Diagram 3) Ordered Pairs (Sets)

1) Arrow Diagram

Considering the hobbies of Mr. Budi’s children, there is a relationship between Set A and Set B. The relationship is said to be “likes to play” from Mr Budi’s children to the types of sports.Riska likes to play badminton and swimmingDimas likes to play FootballCandra likes to play FootballDira likes to play badminton and Basket ballReni likes to play badminton and Basket ball If the expression “likes to play” is represented with an arrow, the statements above can be shown as

Cartesian Diagram

Riska Dimas Candra Dira Reni

2

3 4

5 7

6 8

A

1

Figure 2.3

Riska likes to play badminton and swimmingDimas likes to play FootballCandra likes to play FootballDira likes to play badminton and Basket ballReni likes to play badminton and Basket ball

3) Ordered Pairs (Sets)

Riska likes to play badminton and swimmingDimas likes to play FootballCandra likes to play FootballDira likes to play badminton and Basket ballReni likes to play badminton and Basket ball

In the relationship of likes to play above, we have a set of sports player A = {Riska, Dimas, Candra, Dira, Reni}, and a set of types of sports B = {Badminton, Swimming, Basket ball, Football}. the relation likes to play can be expressed as

R = {(Riska, Swimming), (Riska, Badminton), (Dimas, Football), (Candra, Football), (Dira, Badminton), (Dira, Basket ball),(Reni, Badminton), (Reni, Basket ball)}.

Example

Given A = {3, 4, 5} , B = {1, 2, 5, 7} and “less than” is the relation that connect the A to B.

Express those relations by:

1. Arrow Diagram

2. Cartesian Diagram

3. Ordered Pairs (Sets)

Answer

1) Arrow Diagram

A B

“A less than B”345

1 2 5 7

2) Cartesian Diagram

7

5

2

1

0 3 4 5

B

A

3) Ordered Pairs

{(3, 5), (3, 7), (4, 5), (4, 7), (5, 7)}

Domain, Co domain, Range

Arrow Diagram

A B

345

1 2 5 7

Domain, Co domain, Range

Domain= {3,4,5}

Co domain = {1,2,5,7}

Range = {5,7}

2) Domain: Co domain

A={-1,0,1,2}

Co domain:

B={1,2,3,4,5}

Range:

{1,2,3}

3) Notation of Function:

f:x ax+b

shadow of x

domain

name of function

Function

1. The function of set A to set B is the relationship which relate every member of set A to exactly one member of set B

Example:

Range-1012

1 2 3 4 5

4) The function is expressed by formula:

f(x)=ax+b

The shadow of x by function f written f(x) The shadow of a by function f written f(a)

Cartesian Product

Cartesian product of A and B is expressed by A x B,:

A x B={a,b) | a € A, b € B} n (A x B) = n(A) x n(B)

Example:

If A={p,q} and B={3,5,7}

Determine: a) A x B

b) n(A x B)

Answer

a) A x B = {(p,3), (p.5), (p,7), (q,3), (q,5), (q,7)}

b) n(A x B) = n(A) x n(B) = 2 x 3 = 6

The number of Function of two Sets

If the number of element sets A is n(A) = a and the number of element sets B is n(B) = b, so:

a) The number of the possible function of sets A to B = (b)a

b) The number of the possible function of sets B to A = (a)b

Example

Given A = {4,5,6} and B = {3,5}Determine the number of the possible function of:

a) A to B

b) B to A

Answer

A = {4,5,6} n(A) = a = 3

B = {3,5} n(B) = b = 2

So:

a) The number of function of A to B

= (b)a = 23 = 8

b) The number of function of B to A

= (a)b = 32 = 9

Correspondence One to One

1) Definition :

Correspondence one to one of sets A and sets B is the relationship which relates every member of set A to exactly one member of set B and relates every member of set B to exactly one member of set A.

The number of elements sets A and sets B are equal

2) The number of correspondence one to one

If n(A)=n(B)=n,the number of possible correspondence one to one A and B is;

n x (n-1) x (n-2) x….3x2x1

example

How many the number of corraspondence one to one between sets P and Q, if P = {a, b, c, d} and Q = {3, 5, 7, 9}

answer

n(P) = 4 and n(Q) = 4

The number of correspondence one to one P and Q

= 4 x 3 x 2 x 1

= 24 ways