foundations algebra

Interval Notation and Set Notation

Algebra I20 minDifficulty

You should know: inequalities, domain and range

Overview

Interval notation provides a compact way to write sets of real numbers. (a, b) denotes all reals strictly between a and b; [a, b] includes the endpoints. Mixed parentheses/brackets handle half-open intervals. Infinity is always written with a parenthesis. Set-builder notation {x | condition} is an alternative.

Intuition

Think of intervals on a number line. A bracket [ or ] means 'include the endpoint' (like <= or >=). A parenthesis ( or ) means 'exclude the endpoint' (like < or >). Infinity is always excluded (you can never reach infinity). So [2, 5) means: start at 2 (include it), go up to 5 (but not 5 itself).

Formal Definition

Definition

Open interval: (a, b) = {x in R | a < x < b}. Closed interval: [a, b] = {x | a <= x <= b}. Half-open: [a, b) = {x | a <= x < b}, (a, b] = {x | a < x <= b}. Unbounded: [a, inf) = {x | x >= a}, (-inf, b] = {x | x <= b}, (-inf, inf) = R. Union: A union B written A U B.

(a,b) = \{x \in \mathbb{R} \mid a < x < b\}

Open interval

[a,b] = \{x \in \mathbb{R} \mid a \leq x \leq b\}

Closed interval

[a, \infty) = \{x \in \mathbb{R} \mid x \geq a\}

Unbounded interval

Notation

Notation	Meaning
$(a,b)$	Open interval: a < x < b
$[a,b]$	Closed interval: a <= x <= b
$\mathbb{R}$	All real numbers = (-inf, inf)

Theorems

Theorem 1: Interval as Connected Set

A subset S of R is an interval if and only if it is connected: for all a, b in S with a < b, every point between a and b is also in S.

Worked Examples

1
-3 is included (>=), 7 is excluded (<).
$[-3, 7)$

✓ Answer

[-3, 7)

Practice Problems

Easyfill in blank

Write x <= 5 in interval notation.

Common Mistakes

Common Mistake

Writing (3, inf] with a bracket at infinity.

Infinity is not a real number; it can never be 'reached' or 'included.' Always use a parenthesis: (3, inf).

Quiz

Which interval notation represents -2 < x <= 6?

[-2, 6](-2, 6][-2, 6)(-2, 6)

Historical Background

The notation for intervals was standardized in the 20th century. Georg Cantor's development of set theory (1870s) provided the framework for rigorous treatment of subsets of the reals. Interval notation became standard in analysis and is now universal in algebra through calculus.

1870s
Cantor develops set theory, enabling rigorous treatment of real number subsets
Georg Cantor

Summary

( ) = open endpoint (strictly less/greater). [ ] = closed (less/greater or equal).
Infinity always uses a parenthesis: (a, inf) or (-inf, b].
Union: (1,3) U [5,7) means all x in (1,3) or in [5,7).
Set-builder: {x | a < x <= b} is the same as (a, b].

References

BookHall, B. and Fabricant, M. Algebra 1. Prentice Hall, 2001.