Let A = {a, b c}, B = {u, v, w} and let f and g be two functions from A to B and from B to A respectively defined as: f = {(a, v), (b, u), (c, w)}, g = {(u, b), (v, a), (w, c)}.
Show that f and g both are bijections and find fog and gof.
Given, A = {a, b, c}, B = {u, v, w} and
f = A → B and g: B → A defined by
f = {(a, v), (b, u), (c, w)} and
g = {(u, b), (v, a), (w, c)}
For both f and g, different elements of domain have different
images
∴ f and g are one – one
Again, for each element in co – domain of f and g, there is a pre – image in the domain
∴ f and g are onto
Thus, f and g are bijective.
Now,
gof = {(a, a), (b, b), (c, c.)} and
fog = {(u, u), (v, v), (w, w)}