Let’s refresh the relevant definition we need to know to solve this exercise.
“A function f from A to B is called onto, or a surjection, if and only if for every element b∈B there is an element a∈A with f (a)=b. A function f is called surjective if it is onto.” Discrete Mathematics and its Applications by Rosen.
The functions are defined as follows, f: {a, b, c, d} -> {a, b, c, d}
a) f(a)=b,f(b)=a,f(c)=c,f(d)=d
Notice that for every element of the range y∈{a, b, c, d}, there is an element x such that f(x)=y.
Therefore, the function is onto because it holds the definition of an onto function.
b) f(a)=b,f(b)=b,f(c)=d,f(d)=c
There is no element of the domain associated with the image a.
In other words, there is an element of the range y=a, for which there is no element x from the domain such that f(x)=a.
Therefore, this function is not onto.
c) f(a)=d,f(b)=b,f(c)=c,f(d)=d
There is no element of the domain associated with the image a.
In other words, there is an element of the range y=a, for which there is no element x from the domain such that f(x)=a.
Therefore, this function is not onto.
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