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Work with sets and interval notation

Start a set of 8
Step 4 of 5

Use the set sizes and the shared-element count to find the number of elements in the union

\[\left|A\cup B\right|=\left|A\right|+\left|B\right|-\left|A\cap B\right|=3+3-1=?\]
Year 12, VCE Units 3 & 4, Chapter 2, 4 difficulty levels, No calculatorBuild a worksheet

Sets and interval notation

Use the relation or operation named in the question, and preserve the stated universe and domain. Set notation records exact elements, while interval notation records bounds and whether endpoints are included

\[A\cup B=\left\{x:x\in A\text{ or }x\in B\right\},\quad A\cap B=\left\{x:x\in A\text{ and }x\in B\right\}\]
  1. Match the relation to the objects Membership compares one object with a set, while subset notation compares one set with another set \(\displaystyle 3\in A,\quad A\subseteq B\)
  2. Keep either or both elements A union keeps elements in at least one set, an intersection keeps elements in both sets, and a difference starts with the first set \(\displaystyle \left\{1,4\right\}\cup\left\{4,7\right\}=\left\{1,4,7\right\},\quad\left\{1,4\right\}\cap\left\{4,7\right\}=\left\{4\right\}\)
  3. Use the stated universe A complement contains the elements in the stated universe that are not in the given set \(\displaystyle U=\left\{1,2,3,4\right\},\quad A=\left\{1,3\right\}\Rightarrow U\setminus A=\left\{2,4\right\}\)
  4. Preserve endpoint and domain rules Strict inequalities use open endpoints, inclusive inequalities use closed endpoints, infinity is open, and an integer restriction lists only integers \(\displaystyle -2<x\le4\Longleftrightarrow\left(-2,4\right]\)

Worked example

Write the union of the two finite sets in ascending order

\[A=\left\{-4,1,5\right\},\quad B=\left\{1,3,8\right\}\]
  1. \(\displaystyle A\cap B=\left\{1\right\}\)The shared element is 1.
  2. \(\displaystyle A\cup B=\left\{-4,1,3,5,8\right\}\)Keep every distinct element from either set and write the result in ascending order
  3. \(\displaystyle \left|A\cup B\right|=5\)The union contains five distinct elements

Common mistakes

Union and intersection swapped
✕\(\displaystyle A=\left\{-3,1,4\right\},\quad B=\left\{1,2,6\right\}\Rightarrow A\cup B=\left\{1\right\}\)✓\(\displaystyle A\cup B=\left\{-3,1,2,4,6\right\}\)
A union keeps each distinct element in at least one of the two sets, whereas an intersection keeps only shared elements
Complement ignores the universe
✕\(\displaystyle U=\left\{1,2,3,4\right\},\quad A=\left\{1,3\right\}\Rightarrow U\setminus A=\mathbb{R}\setminus A\)✓\(\displaystyle U=\left\{1,2,3,4\right\},\quad A=\left\{1,3\right\}\Rightarrow U\setminus A=\left\{2,4\right\}\)
A complement is formed inside the universe stated in the question, not automatically inside the real numbers
Open and closed endpoints
✕\(\displaystyle -2<x\le4\Longleftrightarrow\left[-2,4\right)\)✓\(\displaystyle -2<x\le4\Longleftrightarrow\left(-2,4\right]\)
A strict inequality has an open endpoint and an inclusive inequality has a closed endpoint

Difficulty levels

introDecide membership, subset, union or intersection for finite sets
standardForm a complement in a finite universe or convert one inequality to interval notation
challengeForm a set difference, disjoint intervals or an integer-restricted interval
examChoose the only notation that preserves every set, endpoint and domain condition

Questions about this topic

What is the difference between membership and subset notation?+

The membership symbol ∈ relates one object to a set, such as 3 ∈ A. The subset-or-equal symbol ⊆ relates one set to another, such as A ⊆ B.

How does a complement depend on its universe?+

The complement contains only the elements in the stated universe that are not in the given set. Changing the universe can change the complement

When is an interval endpoint open or closed?+

Use an open endpoint for a strict inequality and a closed endpoint for an inclusive inequality. An endpoint at infinity is always open

Can I use a calculator?+

No. This topic is practised without a calculator

Which year level and chapter is this?+

Year 12, VCE Units 3 & 4, Mathematical Methods, chapter 2: Functions and graphs

In this chapter

Read restricted graphs and horizontal level setsFind the gradient of a line through two pointsFind the equation of a line through two pointsSolve a pair of simultaneous linear equationsRead the vertical-axis intercept and gradient from an equationRead the point of intersection of two linesConstruct a linear model from a descriptionInterpret the graph of a linear modelTransform a quadratic and find its key featuresDetermine a quadratic rule from exact informationAnalyse signs and solve a quadratic inequalityAnalyse a cubic graph or determine its ruleInterpret and determine a transformed power functionConnect a factorised higher-degree rule and its graphSolve a higher-degree polynomial equation or inequalityApproximate solutions from intersections of two graphsClassify a relation and state its domain and rangeEvaluate and determine function rulesApply and describe transformations of a functionAnalyse the general rational-power familyFind the maximal domain of an algebraic functionAnalyse reciprocal and square-reciprocal functionsAnalyse sideways-parabola and root-function familiesDetermine a power-family rule from features and pointsWork with sets and interval notationSimplify, compare and operate with surdsCombine function rules and ordinatesRecognise equivalent polynomial formsUse a piecewise linear modelEvaluate exact circular-function values by symmetry and periodicityKey points and features of the basic sine and cosine graphsTransform and reconstruct sine and cosine functionsFeatures and transformations of tangent functionsForm and use periodic sine and cosine modelsAnalyse exponential and logarithmic graphsInterpret and use an exponential modelSelect and interpret a function modelEvaluate and interpret a restricted rational model