1. ## Discrete maths

stuck on this one

2. Originally Posted by sabina_19
stuck on this one

a) (fog)(2)=f(g(2))=f(2*2)=f(4)=4+1=5

b) (gof)(2)=g(f(2)) = g(2+1) = g(3) = 2*3=6

Hence $(fog)(2)\neq (gof)(2)$

The function: f={(1,a),(2,a),(4,c),(3,d) from A={1,2,3,4} TO B= {a,b,c,d}
IS not one to one and onto,since

there exist x=1 ,y=2, such that f(x) =f(y)=a, and $x\neq y$

Note for a function to be one to one we must have:

for all x, yεA if f(x)=f(y) ,then x=y

Also for a function to be onto we must have:

for all y belonging to B ,yεB, there exists xεA SUCH that f(x)=y.

However in our case we note that:

there exists y=bεB SUCH that FOR no element of xεA WE have f(x)=y=b

Hence there is no inverse function from B TO A.

This can be seen in another way:

Let the inverse of f be defined as :f^(-1)={(y,x): (x,y)εf},and according to that definition :

f^(-1) = {(a,1),(a,2),(c,4),(d,3)}, and using the definition of a function we observe that f^(-1) ( the inverse of f) cannot be a function from B TO A