Functions and Onto (Computer Science Notes)
Functions
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We’ll be using a broader range of functions, whose input and/or output values may be
integers, strings, characters, and the like.
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Suppose that A and B are sets, then a function f from A to B (shorthand: f : A → B) is an
assignment of exactly one element of B (i.e. the output value) to each element of A (i.e.
the input value)
•
A is called the domain of f and B is called the co-domain
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If x is an element of A, then the value f(x) is also known as the image of x, output is an
image
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We can write out the elements of A as x1, x2, . . . , xn. When constructing a function f : A
→ B, we have p ways to choose the output value for x1
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The choice of f(x1) doesn’t affect our possible choices for f(x2): we also have p choices
for that value
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we have p^n possible ways to construct our function f
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For any set A, the
identity function
idA maps each value in A to itself. That is, idA : A →
A and idA(x) = x
When Are Functions Equal?
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To be equal, two functions must have the same domain, the same co-domain, and assign
the same output value to each input value
•
• f: N → N such that f(x) = 2x.
• f: R → R such that f(x) = 2x Describe quite different functions, even though they are
based on the same equation
What Isn’t A Function?
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For each input value, a function must provide one and only one output value, so if there
exist an input with no output it’s not a function or if one input has two outputs it’s not a
function
Images and Onto
•
The image of the function f: A → B is the set of values produced when f is applied to all
elements of A. That is, the image is f(A) = {f(x) : x
∈
A}
•
For example, suppose M = {a, b, c, d}, N = {1, 2, 3, 4}, and our function g: M → N is as
in the following diagram. Then g(A) = {1, 3, 4}
•
A
1
B
1
2
C
3
D
4
•
A function f : A → B is onto (or subjective) if its image is its whole co-domain
∀
y
∈
B,
∃
x
∈
A, f(x) = y
•
Whether a function is onto critically depends on what sets we’ve picked for its domain