The addition and multiplication Solution Definitions: A function f is one-to-one if f maps distinct elements in the domain to distinct elements in the range. This is equivalent to saying that f(x)=f(y) implies x=y, as otherwise you\'d have distinct x,y that mapped to the same value f(x),f(y). A function f is onto if for every y in the range of f, there is some element x in the domain of f such that f(x)=y. In other words, every element in the range gets mapped to. This problem has a Cartesian product as the domain, so every element of its domain can be thought of as an ordered pair (x,y). We examine mult(x,y) = xy. Mult is not one-to-one. There are many elements in the range that are mapped to by more than one ordered pair in the domain. Here is a counterexample: mult(4,6) = 4*6 = 24 mult(3,8) = 3*8 = 24 As you can see, any pair of factors of a number will map to that number, so mult does not map distinct ordered pairs in the domain to distinct numbers in the range, and is therefore not one-to- one. Mult is, however, onto. Consider an element y in R. 1*y = y. So the ordered pair (1,y) will be mapped to y by mult: mult(1,y) = y As y is in R because it is in the range and 1 is a real number, (1,y) is in RxR. Therefore, for any element in the range of mult there is some element in the domain that maps to that element, and mult is onto..

1. The addition and multiplication
Solution
Definitions:
A function f is one-to-one if f maps distinct elements in the domain to distinct elements in the
range. This is equivalent to saying that f(x)=f(y) implies x=y, as otherwise you'd have distinct
x,y that mapped to the same value f(x),f(y).
A function f is onto if for every y in the range of f, there is some element x in the domain of f
such that f(x)=y. In other words, every element in the range gets mapped to.
This problem has a Cartesian product as the domain, so every element of its domain can be
thought of as an ordered pair (x,y).
We examine mult(x,y) = xy.
Mult is not one-to-one. There are many elements in the range that are mapped to by more than
one ordered pair in the domain. Here is a counterexample:
mult(4,6) = 4*6 = 24
mult(3,8) = 3*8 = 24
As you can see, any pair of factors of a number will map to that number, so mult does not map
distinct ordered pairs in the domain to distinct numbers in the range, and is therefore not one-to-
one.
Mult is, however, onto. Consider an element y in R. 1*y = y. So the ordered pair (1,y) will be
mapped to y by mult:
mult(1,y) = y
As y is in R because it is in the range and 1 is a real number, (1,y) is in RxR. Therefore, for any
element in the range of mult there is some element in the domain that maps to that element, and
mult is onto.