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Beginning Activity Beginning Activity 1: Functions and Sets
A4US

Let \(S = \left\{ a, b, c, d \right\}\) and \(T = \left\{ s, t, u \right\}\text{.}\) Define \(f\x S \to T\) by

\begin{align*} f(a) \amp = s \amp f(b) \amp = t \amp f(c) \amp = t \amp f(d) \amp = s\text{.} \end{align*}

1.

Let \(A = \left\{ a,c \right\}\) and \(B = \left\{ a, d \right\}\text{.}\) Notice that \(A\) and \(B\) are subsets of \(S\text{.}\) Use the roster method to specify the elements of the following two subsets of \(T\text{:}\)

(a)

\(\left\{ f ( x ) \mid x \in A \right\}\)

(b)

\(\left\{ f ( x ) \mid x \in B \right\}\)

2.

Let \(C = \left\{ s, t \right\}\) and \(D = \left\{ s, u \right\}\text{.}\) Notice that \(C\) and \(D\) are subsets of \(T\text{.}\) Use the roster method to specify the elements of the following two subsets of \(S\text{:}\)

(a)

\(\left\{ x \in S \mid f ( x ) \in C \right\}\)

(b)

\(\left\{ x \in S \mid f ( x ) \in D \right\}\)

Now let \(g\x \R \to \R\) be defined by \(g ( x ) = x^2\text{,}\) for each \(x \in \mathbb{R}\text{.}\)

3.

Let \(A = \left\{ 1, 2, 3, -1 \right\}\text{.}\) Use the roster method to specify the elements of the set \(\left\{ g ( x ) \mid x \in A \right\}\text{.}\)

4.

Use the roster method to specify the elements of each of the following sets:

(a)

\(\left\{ x \in \R \mid g( x ) = 1 \right\}\)

(b)

\(\left\{ x \in \R \mid g( x ) = 9 \right\}\)

(c)

\(\left\{ x \in \R \mid g( x ) = 15 \right\}\)

(d)

\(\left\{ x \in \R \mid g( x ) = -1 \right\}\)

5.

Let \(B = \left\{ 1, 9, 15, -1 \right\}\text{.}\) Use the roster method to specify the elements of the set \(\left\{ x \in \mathbb{R} \mid g ( x ) \in B \right\}\text{.}\)