198 CHAPTER 4 Abstract Algebra
If f :A→B is a mapping we callAthedomainof f and callB
thecodomainoff. Therangeoff is the subset{f(a)|a∈A}⊆B.
Some definitions. Let AandB be sets and letf :A→B. We say
that
f is one-to-one(or isinjective) if wheneverx, y∈A, x 6 =y then
f(x) 6 =f(y). (This is equivalent with saying thatf(x) =f(y)⇒
x=y, where x, y∈A.)
fisonto(or issurjective) if for anyz∈Bthere is an elementx∈A
such thatf(x) =z. Put differently, f is onto if the range ofris
all ofB.
f isbijectiveiff is both one-to-one and onto.The following definition is extremely useful. Let Aand B be sets
and letf : A→B be a mapping. Let b∈B; the fibreof f over b,
writtenf−^1 (b) is the set
f−^1 (b) ={a∈A|f(a) =b}⊆A.Please do not confuse fibres with anything having to do with
the inverse function f−^1 , as this might not exist! Note that if
b ∈B then the fibre over b might be the empty set. However, if we
know thatf :A→B is onto, then the fibre over each element ofBis
nonempty. If, in fact, for eachb∈B the fibref−^1 (b) overbconsists of
a single element, then we are guaranteed thatf is a bijection.
Finally, a mappingf :A→Afrom a set into itself is called apermu-
tationif it is a bijection. It should be clear that if|A|=n, then there
aren! bijections onA.
Exercises
- Letf:R→Rbe aquadraticfunction. Therefore, there are real
constantsa, b, c∈Rwitha 6 = 0 such thatf(x) =ax^2 +bx+cfor
allx∈R. Prove thatf cannot be either injective or surjective.