ONTO ONE TO ONE FUNCTION.ppt

ONTO ONE TO ONE FUNCTION

1. BS SE(1B)EVENING
COURSE:DISCRETESTRUCTURE
UIIT
GROUP NO:8
GROUP MEMBERS :
1.SAJAL ASGHAR
2.M.MUAWWAZ USMAN
3.M.SAAD MALIK
UIIT(UNIVERSITY INSTITUTEOF INFORMATION TECHNOLOGY )

2. OUTLINE OF TODAY’S
PRESENTTATION:
ONE-ONE FUNCTION.
ONTO FUNCTION
BIJECTIVE FUNCTION.

3. ONE-ONE
FUNCTION(INJECTIVE)
DEFINATION:-
“A function of from set ‘A’ to set ‘B’ is one-one
Function if no two elements in ‘A’ are mapped
To same element in ‘B’ ”
Distinct elements form distinct images.
We cannot mapped with same elements.
Value of ‘x’ elements must be equal to ’y’.
Value of ‘y’ must be greater than ‘x’.

4. EXAMPLES:-
1
2
3
A
B
c
1
2
3
4
A
B
d
1
2
3
A
B
C
d
1
2
3
A
C
b

5. |A|<|B|
How many one to one functions are possible?
Total number of element in set ‘A’ is |x|.
Total number of elements in set ‘B’ is |Y|.
I have ‘y’ number of choices for first element of set ‘A’.
{y(y-1) (y-2) (y-3)…….y-(x-1)}
So we can derive formula from this series.
 yPx =4P3 this is how to write formula
1
2
3
A
C
B
k

6. ONTO FUNCTION (surjection)
DEFINATION:-
‘A function ‘f’ from set ‘A’ to set ‘B’ is onto if each element of
’B’ is mapped to at least one element of ‘A’.
Range of f=B.
In onto function range equals to the elements of set B.
If both range and co-domains are equal so it is onto
function.
|A|>|B|
A
B
C
1
2
3
1
2
3
A
B

7. If one-one then onto?
If onto then one-one?
How many onto functions if |A|=|B|?
1
2
3
A
B
1
2
3
A
B
C
d
1
2
3
A
B
c

8. BIJECTIVE FUNCTION:
DEFINATION:-
“A Function f : A&B bijection if ‘f’is both one-one & onto.”
One-one function:-( |A| <|B| )
Onto function:-( |A| >|B| )
If you want to make bijective function then number of
element of set ‘A’ must be equal to the elements of set of
set ‘B’. |A|=|B|

9. EXAMPLES:-
1.First is one-one but not onto.
2.Second is onto but not one-one.
3.Third is one-one & onto.
1
2
3
4
A
B
c
1
2
3
A
B
C
d
1
2
3
A
B
c