Beginning Activity Beginning Activity 2: Statements Involving Functions
Let \(A\) and \(B\) be nonempty sets and let \(f\x A \to B\text{.}\) In Beginning Activity 1, we determined whether or not certain functions satisfied some specified properties. These properties were written in the form of statements, and we will now examine these statements in more detail.
1.
Consider the following statement:
For all \(x, y \in A\text{,}\) if \(x \ne y\text{,}\) then \(f ( x ) \ne f ( y )\text{.}\)
(a)
Write the contrapositive of this conditional statement.
(b)
Write the negation of this conditional statement.
2.
Now consider the statement:
For all \(y \in B\text{,}\) there exists an \(x \in A\) such that \(f ( x ) = y\text{.}\)Write the negation of this statement.
3.
Let \(g:\R \to \R\) be defined by \(g ( x ) = 5x + 3\text{,}\) for all \(x \in \R\text{.}\) Complete the following proofs of the following propositions about the function \(g\text{.}\)
- Propostion 1
For all \(a, b \in \R\text{,}\) if \(g ( a ) = g ( b )\text{,}\) then \(a = b\text{.}\)
- Proof
-
We let \(a, b \in \R\text{,}\) and we assume that \(g ( a ) = g ( b )\) and will prove that \(a = b\text{.}\) Since \(g(a) = g(b)\text{,}\) we know that
\begin{equation*} 5a + 3 = 5b + 3\text{.} \end{equation*}(Now prove that in this situation, \(a = b\text{.}\))
- Proposition 2
For all \(b \in \R\text{,}\) there exists an \(a \in \R\) such that \(g ( a ) = b\text{.}\)
- Proof
-
We let \(b \in \R\text{.}\) We will prove that there exists an \(a \in \R\) such that \(g ( a ) = b\) by constructing such an \(a\) in \(\R\text{.}\) In order for this to happen, we need \(g(a) = 5a + 3 = b\text{.}\)
(Now solve the equation for \(a\) and then show that for this real number \(a\text{,}\) \(g ( a ) = b\text{.}\))