 Solution

ID: past-exam-of-the-year-1-of-the-computer-science-course-of-the-university-of-oxford/2016/discrete-mathematics-continuous-mathematics-and-probability-resit/1/c/solution

f : A \to B

is onto, or a surjective function, when every

b \in B

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

. It is one-to-one, or an injective function, when

f (a) = f (a^{'})

implies

a = a^{'}

for all

a, a^{'} \in A

. It is bijective, a bijection, when both properties hold.

f^{- 1} : B \to A

satisfying

f^{- 1} \circ f = id_{A}, f \circ f^{- 1} = id_{B} .

f^{- 1} (b)

is the unique

a

with

f (a) = b

: existence follows from surjectivity and uniqueness from injectivity. This is a two-sided inverse function, rather than just a left or right inverse.

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