Past exam of the year 1 of the computer science course of the University of Oxford / 2013 / Discrete mathematics, probability and continuous mathematics / 1 / d

Table of contents
- i d
  - Solution i
- ii d
  - Solution ii
- iii d
  - Solution iii
- iv d
  - Solution iv

i

Solution

The image of a function is the set of values actually attained as its argument ranges over its domain of a function:

im f = f (A) = {f (a) : a \in A} = {b \in B : \exists a \in A, f (a) = b} .

It is a subset of the codomain

B

, and need not equal

B

.

ii

Solution

The function is surjective, also called onto, when every element of its codomain has at least one preimage:

f (A) = B ⟺ \forall b \in B \exists a \in A : f (a) = b .

The condition concerns the specified codomain, not just the formula defining the function.

iii

Solution

The function is injective, also called one-to-one, when equal outputs force equal inputs:

\forall a, a^{'} \in A, f (a) = f (a^{'}) ⟹ a = a^{'} .

Equivalently, different elements of the domain of a function always have different images.

iv

Solution

A bijection is a function that is both injective and surjective. Each element of the codomain then has exactly one preimage:

\forall b \in B \exists! a \in A : f (a) = b .

This unique-preimage property allows the inverse function

f^{- 1} : B \to A

to be defined on all of

B

.

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