Sort each description by the property of a function it is describing.
Groups: A statement about injectivity · A statement about surjectivity
- If f(a) equals f(b) then a equals b
- There are no collisions between inputs
- No two different inputs share an output
- Nothing in the codomain is left uncovered
- Every element of the codomain is hit by some input
- The image is the whole of the codomain
Hints
- Ask whether the description constrains the inputs or the codomain.
- Collisions are about the domain side; coverage is about the codomain side.
Show the answer
A statement about injectivity: No two different inputs share an output, If f(a) equals f(b) then a equals b, There are no collisions between inputs
A statement about surjectivity: Every element of the codomain is hit by some input, The image is the whole of the codomain, Nothing in the codomain is left uncovered
Injectivity restricts the inputs, distinct inputs must give distinct outputs, so nothing collides. Surjectivity restricts the codomain, nothing may be left out, so the image fills it. A function can have either without the other, and a bijection is precisely one that has both, which is what makes it invertible.
Practise Functions
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