Question

(a) Prove that if $n \in \mathbb{N}$ and $A \subseteq\left\{i \in \mathbb{Z}^{+} \mid i \leq n\right\}$, then $A$ is finite and $A$ has at most $n$ elements. Furthermore, if $A \neq\left\{i \in \mathbb{Z}^{+} \mid i \leq n\right\}$, then $A$ has fewer than $n$ elements. (b) Suppose $B \subseteq A, B \neq A$, and $B \sim A$. Prove that $A$ is infinite.

   (a) Prove that if $n \in \mathbb{N}$ and $A \subseteq\left\{i \in \mathbb{Z}^{+} \mid i \leq n\right\}$, then $A$ is finite and $A$ has at most $n$ elements. Furthermore, if $A \neq\left\{i \in \mathbb{Z}^{+} \mid i \leq n\right\}$, then $A$ has fewer than $n$ elements.
(b) Suppose $B \subseteq A, B \neq A$, and $B \sim A$. Prove that $A$ is infinite.

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How to Prove It: A Structured Approach

How to Prove It: A Structured Approach

Daniel J. Velleman 1st Edition

Chapter 7, Problem 7

Instant Answer

Step 1

Base case: When $n = 1$, the set $\{i \in \mathbb{Z}^{+} \mid i \leq n\}$ is $\{1\}$. Any subset $A$ of this set can either be $\{1\}$ or the empty set $\{\}$. In either case, $A$ is finite and has at most $1$ element. Inductive step: Assume that for some $k \in Show more…

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(a) Prove that if $n \in \mathbb{N}$ and $A \subseteq\left\{i \in \mathbb{Z}^{+} \mid i \leq n\right\}$, then $A$ is finite and $A$ has at most $n$ elements. Furthermore, if $A \neq\left\{i \in \mathbb{Z}^{+} \mid i \leq n\right\}$, then $A$ has fewer than $n$ elements. (b) Suppose $B \subseteq A, B \neq A$, and $B \sim A$. Prove that $A$ is infinite.

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