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Math Help - Discrete maths

  1. #1
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    Discrete maths

    stuck on this one
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  2. #2
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    Quote Originally Posted by sabina_19 View Post
    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
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