Sets of Numbers

Context: FIT1058_MOC · the named standard sets used throughout computing · restricted by sign/zero · refined into intervals

Quick Revision

• 🎯 Objective: reserved symbols for common number sets ➔ each carries its usual operations.
• 📦 Core Components: $\mathbb{N}\subset \mathbb{Z}\subset \mathbb{Q}\subset \mathbb{R}$ ➔ sign/zero restrictions ➔ interval notation.
• ⚡ Key Constraint: $\mathbb{N}$ vs ${\mathbb{N}}_0$ differ only by zero; bracket type fixes interval endpoints.

📝 Core

1. The Named Sets

• $\mathbb{N}=\{1,2,3,\dots \}$ ➔ positive integers; ${\mathbb{N}}_0$ ➔ adds $0$.
• $\mathbb{Z}$ ➔ all integers; $\mathbb{Q}=\{p/q:p,q\in \mathbb{Z},q\ne 0\}$ ➔ rationals; $\mathbb{R}$ ➔ reals.
• Chain ➔ $\mathbb{N}\subset \mathbb{Z}\subset \mathbb{Q}\subset \mathbb{R}$ (proper subsets).

2. Sign/Zero Restrictions

• Superscript = sign ➔ ${\mathbb{Z}}^{+}=\mathbb{N}$, ${\mathbb{R}}^{-}=\{x<0\}$.
• Subscript $0$ = “and zero” ➔ ${\mathbb{R}}_0^{+}=\{x\ge 0\}$, ${\mathbb{Q}}_0^{-}=\{x\le 0\}$.

3. Interval Notation

• Square = includes endpoint; round = excludes.
• Forms ➔ $[a,b]$ closed, $[a,b)$/$(a,b]$ half-open, $(a,b)$ open.
• Subscript ➔ $[a,b{]}_{\mathbb{Z}}$ = integers between $a,b$ inclusive.

Key identities:

$\mathbb{N}=\{1,2,\dots \},{\mathbb{N}}_0=\{0,1,\dots \},\mathbb{Z},\mathbb{Q}=\{p/q:q\ne 0\},\mathbb{R}$ $[a,b]=\{x\in \mathbb{R}:a\le x\le b\},[a,b)=\{a\le x<b\},(a,b)=\{a<x<b\}$

⚖️ Core Decision Matrix

Symbol	Set	Note
$\mathbb{N}$	$\{1,2,3,\dots \}$	positive integers
${\mathbb{N}}_0$	$\{0,1,2,\dots \}$	adds zero
${\mathbb{Z}}^{+}$	$\mathbb{N}$	superscript = sign
${\mathbb{R}}_0^{+}$	$\{x\ge 0\}$	positives + zero

When It Flips: each name denotes the set and its operations — context decides whether "$\mathbb{Z}$" means the elements or the system $(\mathbb{Z},+,\times )$. A number set (e.g. $\mathbb{Z}$) often serves as the universal set $U$ for a numeric problem.

📊 Exam Execution Trace

Manual Execution Trace

Smallest standard set each value belongs to:

Step / State	Value	Smallest set
0 (Init)	—	—
1	$-3$	$\mathbb{Z}$
2	$0$	${\mathbb{N}}_0$
3	$\frac{2}{5}$	$\mathbb{Q}$
4	$\sqrt{2},\pi$	$\mathbb{R}$ (irrational)

⚠️ Common Mistakes

• 💡 $\mathbb{N}$ vs ${\mathbb{N}}_0$ ➔ differ only by $0$ ($0\in {\mathbb{N}}_0$, $0\notin \mathbb{N}$ in this unit); a square bracket includes its endpoint, a round bracket excludes it.

🧠 Active Recall

Distinguish $\mathbb{N}$, ${\mathbb{N}}_0$, ${\mathbb{Z}}^{+}$, and ${\mathbb{R}}_0^{+}$.

• Hint: Sign superscript, zero subscript.

Answer

• Short answer: $\mathbb{N}=\{1,2,\dots \}$; ${\mathbb{N}}_0$ adds $0$; ${\mathbb{Z}}^{+}=\mathbb{N}$; ${\mathbb{R}}_0^{+}=\{x\ge 0\}$.
• Why: Notation ➔ superscript sets the sign, subscript $0$ re-admits zero.

Write $[a,b)$ and $(a,b)$ as set-builder and say which endpoints are members; what is $[a,b{]}_{\mathbb{Z}}$?

• Hint: Bracket = membership.

Answer

• Short answer: $[a,b)=\{a\le x<b\}$ (includes $a$); $(a,b)=\{a<x<b\}$ (open); $[a,b{]}_{\mathbb{Z}}=\{x\in \mathbb{Z}:a\le x\le b\}$.
• Why: Square includes, round excludes ➔ the subscript restricts to integers.