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1
Definition of a function
• A function takes an element from a set and
maps it to a UNIQUE element in another set
2
More functions
1
2
3
4
5
“a”
“bb“
“cccc”
“dd”
“e”
A string length function
A
B
C
D
F
Alice
Bob
Chris
Dave
Emma
A class grade function
Domain Co-domain
A pre-image
of 1
The image
of A
3
One-to-one functions
1
2
3
4
5
a
e
i
o
A one-to-one function
1
2
3
4
5
a
e
i
o
A function that is
not one-to-one
• A function is one-to-one if each element in the
co-domain has a unique pre-image
– Meaning no 2 values map to the same result
4
More on one-to-one
• Injective is synonymous with one-to-one
– “A function is injective”
• A function is an injection if it is one-to-one
• Note that there can
be un-used elements
in the co-domain
1
2
3
4
5
a
e
i
o
A one-to-one function
5
Onto functions
1
2
3
4
5
a
e
i
o
A function that
is not onto
• A function is onto if each element in the co-
domain is an image of some pre-image
– Meaning all elements in the right are mapped to
1
2
3
4
a
e
i
o
u
An onto function
6
1
2
3
4
a
e
i
o
u
An onto function
More on onto
• Surjective is synonymous with onto
– “A function is surjective”
• A function is an surjection if it is onto
• Note that there can
be multiply used
elements in the
co-domain
7
Onto vs. one-to-one
• Are the following functions onto, one-to-one,
both, or neither?
1
2
3
4
a
b
c
1
2
3
a
b
c
d
1
2
3
4
a
b
c
d
1
2
3
4
a
b
c
d
1
2
3
4
a
b
c
1-to-1, not onto
Onto, not 1-to-1
Both 1-to-1 and onto Not a valid function
Neither 1-to-1 nor onto
8
Bijections
• Consider a function that is
both one-to-one and onto:
• Such a function is a one-to-one correspondence,
or a bijection
1
2
3
4
a
b
c
d

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onto into bijective.ppt

  • 1. 1 Definition of a function • A function takes an element from a set and maps it to a UNIQUE element in another set
  • 2. 2 More functions 1 2 3 4 5 “a” “bb“ “cccc” “dd” “e” A string length function A B C D F Alice Bob Chris Dave Emma A class grade function Domain Co-domain A pre-image of 1 The image of A
  • 3. 3 One-to-one functions 1 2 3 4 5 a e i o A one-to-one function 1 2 3 4 5 a e i o A function that is not one-to-one • A function is one-to-one if each element in the co-domain has a unique pre-image – Meaning no 2 values map to the same result
  • 4. 4 More on one-to-one • Injective is synonymous with one-to-one – “A function is injective” • A function is an injection if it is one-to-one • Note that there can be un-used elements in the co-domain 1 2 3 4 5 a e i o A one-to-one function
  • 5. 5 Onto functions 1 2 3 4 5 a e i o A function that is not onto • A function is onto if each element in the co- domain is an image of some pre-image – Meaning all elements in the right are mapped to 1 2 3 4 a e i o u An onto function
  • 6. 6 1 2 3 4 a e i o u An onto function More on onto • Surjective is synonymous with onto – “A function is surjective” • A function is an surjection if it is onto • Note that there can be multiply used elements in the co-domain
  • 7. 7 Onto vs. one-to-one • Are the following functions onto, one-to-one, both, or neither? 1 2 3 4 a b c 1 2 3 a b c d 1 2 3 4 a b c d 1 2 3 4 a b c d 1 2 3 4 a b c 1-to-1, not onto Onto, not 1-to-1 Both 1-to-1 and onto Not a valid function Neither 1-to-1 nor onto
  • 8. 8 Bijections • Consider a function that is both one-to-one and onto: • Such a function is a one-to-one correspondence, or a bijection 1 2 3 4 a b c d