Let X, Y, W be sets and f X Y and g Y W be bijections. Show .pdf

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The document shows that the composition of two bijective functions is also a bijection. It defines three sets X, Y, W and bijective functions f: X->Y and g: Y->W. It then defines a function h: X->W as h(x) = g(f(x)). It proves h is bijective by showing it is both one-to-one and onto. It demonstrates one-to-one by showing each x maps to a unique w, and onto by showing each w maps back to some x. Therefore, the composition of f and g, defined as h, is a bijection from X to W.

Formation

Let X, Y, W be sets and f : X ? Y and g : Y ? W be bijections. Show that h : X ? W defined by
h(x) = g(f(x)) is a bijection
Solution
given f : x -> y is bijection
g : y -> z is bijection
let{x1,x2,x3...xn}belong to X
let{y1,y2,y3...yn}belong to Y
implies let{w1,w2,w3...wn}belong to W
for one one
x1 R y1
thereexists such that
y1 R w1 since g is bijection
implies x1 R w1
hence h is one -one
for onto
consider w1 in w
since g is onto
there exists y1 in Y
such that
y1 R w1
since f is onto for y1 thereexists x1
such that x1 R y1
hence h is onto
hence h is bijection
f(x)=y
g(y)=z
for h(x)=w = g(y) from above
implies h(x)= g(f(x))

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