(3 points) Determine whether each of these functions f : {a, b, c, d} → {a, b, c, d} is one-to-one and whether each of these functions is onto: (a) f(a) = b, f(b) = a, f(c) = c, f(d) = d (b) f(a) = b, f(b) = b, f(c) = d, f(d) = c (c) f(a) = d, f(b) = b, f(c) = c, f(d) = d (2) (3 points) Determine whether each of these functions f : R → R is a one-to-one correspondence (i. e., onto and one-t

Step-by-step explanation:

Recall the following. Given a function f we say that  

f is one to one if for x,y in the domain of f and f(x)=f(y) then x=y.

This means that there are no different elements in the domain of the function  that are linked to the same value.

f is onto if for every y in the codomain of f, then there is an element x in the domain of f such that f(x) = y. That is, given an element y in the codomain of f, there exists and element in the domain that is linked to y.

a)  

f(a) =b, f(b)=a, f(c) = c, f(d) = d. Note that this function links each element in {a,b,c,d} to different elements in {a,b,c,d}, so it is a one to one function. Since for every element x in {a,b,c,d} there is an element y in {a,b,c,d} such that f(y) = x, then this function is also onto.  

b)  

f(a)=b, f(b) = b, f(c) = d, f(d) = c. Note that the elements a,b are linked both to b, so this function fails to be one-to-one. Also, note that the element a has no element x in {a,b,c,d} such that f(x) = a, so this function fails to be onto.  

c)  

f(a) = d, f(b) = b, f(c) = c, f(d) = d.  

As in b), since a and d are linked to d, then this function is not one to one. Also, since the element a has no element x in {a,b,c,d} such that f(x) = a, then it also fails to be onto.  

over the argument its not good to fight.