Beginning Activity Beginning Activity 1: Equivalent Sets, Part 1
1.
Let \(A\) and \(B\) be sets and let \(f\) be a function from \(A\) to \(B\text{.}\) \(\left(f:A \to B \right)\text{.}\) Carefully complete each of the following using appropriate quantifiers: (If necessary, review the material in Section 6.3.)
(a)
The function \(f\) is an injection provided that \(\ldots \text{.}\)
(b)
The function \(f\) is not an injection provided that \(\ldots \text{.}\)
(c)
The function \(f\) is a surjection provided that \(\ldots \text{.}\)
(d)
The function \(f\) is not a surjection provided that \(\ldots \text{.}\)
(e)
The function \(f\) is a bijection provided that \(\ldots \text{.}\)
Definition.
Let \(A\) and \(B\) be sets. The set \(A\) is equivalent to the set \(B\) provided that there exists a bijection from the set \(A\) onto the set \(B\text{.}\) In this case, we write \(A \approx B\text{.}\) When \(A \approx B\text{,}\) we also say that the set \(A\) is in one-to-one correspondence with the set \(B\) and that the set \(A\) has the same cardinality as the set \(B\text{.}\)
Note: When \(A\) is not equivalent to \(B\text{,}\) we write \(A \not \approx B\text{.}\)
2.
For each of the following, use the definition of equivalent sets to determine if the first set is equivalent to the second set.
(a)
\(A = \left\{ 1, 2, 3 \right\}\) and \(B = \left\{ a, b, c \right\}\)
(b)
\(C = \left\{ 1, 2 \right\}\) and \(B = \left\{ a, b, c \right\}\)
(c)
\(X = \left\{ 1, 2, 3, \ldots, 10 \right\}\) and \(Y = \left\{ 57, 58, 59, \ldots, 66 \right\}\)
3.
Let \(D^+\) be the set of all odd natural numbers. Prove that the function \(f\x \mathbb{N} \to D^+\) defined by \(f \left( x \right) = 2x - 1\text{,}\) for all \(x \in \mathbb{N}\text{,}\) is a bijection and hence that \(\mathbb{N} \approx D^+\text{.}\)
4.
Let \(\R^+\) be the set of all positive real numbers. Prove that the function \(g\x \R \to \R^+\) defined by \(g (x ) = e^x\text{,}\) for all \(x \in \R\) is a bijection and hence, that \(\R \approx \R^+\text{.}\)