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Beginning Activity Beginning Activity 1: Functions with Finite Domains
A4US

Let \(A\) and \(B\) be sets. Given a function \(f\x A \to B\text{,}\) we know the following:

  • For every \(x \in A\text{,}\) \(f( x ) \in B\text{.}\) That is, every element of \(A\) is an input for the function \(f\text{.}\) This could also be stated as follows: For each \(x \in A\text{,}\) there exists a \(y \in B\) such that \(y = f( x )\text{.}\)

  • For a given \(x \in A\text{,}\) there is exactly one \(y \in B\) such that \(y = f( x )\text{.}\)

The definition of a function does not require that different inputs produce different outputs. That is, it is possible to have \(x_1 , x_2 \in A\) with \(x_1 \ne x_2\) and \(f( {x_1 } ) = f( {x_2 } )\text{.}\) The arrow diagram for the function \(f\) in FigureĀ 6.15 illustrates such a function.

Also, the definition of a function does not require that the range of the function must equal the codomain. The range is always a subset of the codomain, but these two sets are not required to be equal. That is, if \(g\x A \to B\text{,}\) then it is possible to have a \(y \in B\) such that \(g( x ) \ne y\) for all \(x \in A\text{.}\) The arrow diagram for the function \(g\) in FigureĀ 6.15 illustrates such a function.

Function f has different inputs which do not produce different outputs. The range of function g does not equal the codomain.
Figure 6.15. Arrow Diagram for Two Functions

Now let \(A = \left\{ {1,2,3} \right\}\text{,}\) \(B = \left\{ {a,b,c,d} \right\}\text{,}\) and \(C = \left\{ {s,t} \right\}\text{.}\) Define

\(f\x A \to B\) by \(g\x A \to B\) by \(h\x A \to C\) by
\(f( 1 ) = a\) \(g( 1 ) = a\) \(h( 1 ) = s\)
\(f( 2 ) = b\) \(g( 2 ) = b\) \(h( 2 ) = t\)
\(f( 3 ) = c\) \(g( 3 ) = a\) \(h( 3 ) = s\)

1.

Which of these functions satisfy the following property for a function \(F\text{?}\)

For all \(x, y \in \text{ dom} ( F )\text{,}\) if \(x \ne y\text{,}\) then \(F(x) \ne F(y)\text{.}\)

2.

Which of these functions satisfy the following property for a function \(F\text{?}\)

For all \(x, y \in \text{ dom} ( F )\text{,}\) if \(F( x ) = F( y )\text{,}\) then \(x = y\text{.}\)

3.

Determine the range of each of these functions.

4.

Which of these functions have their range equal to their codomain?

5.

Which of the these functions satisfy the following property for a function \(F\text{?}\)

For all \(y\) in the codomain of \(F\text{,}\) there exists an \(x \in \text{ dom} ( F )\) such that \(F( x ) = y\text{.}\)