Finite sets and their elements
Use exact values, distinguish an object from a set, and preserve the stated universe
- Element and subset Membership compares one object with a set. A subset requires every element of one set to be in the other \(\displaystyle 1\in A,\quad\{1\}\subseteq A\)
- Union and intersection Union keeps every distinct element in either set; intersection keeps only shared elements \(\displaystyle A\cup B,\quad A\cap B\)
- Difference Start with the first set and remove elements in the second. The order matters \(\displaystyle A\setminus B\)
- Complement Start with the stated universe, then remove the set. Do not introduce values outside the universe. The symbol R in double-struck type means all real numbers, not the finite universe \(\displaystyle A'=U\setminus A\)
- Empty and cardinality The empty set has no elements. n(A), N(A) and |A| count distinct elements \(\displaystyle n(\varnothing)=0\)
Worked example
Let A={-2,1,4}, B={1,3,4} and U={-2,0,1,3,4,6}. Find A only, then the complement of the union
\[A\setminus B,\quad(A\cup B)'\]
- \(\displaystyle A\setminus B=\{-2\}\)Remove the shared elements from A.
- \(\displaystyle A\cup B=\{-2,1,3,4\}\)List distinct elements in either set
- \(\displaystyle (A\cup B)'=\{0,6\}\)Keep the universe elements outside that union
Common mistakes
Membership and subset mixed up
✕\(\displaystyle 1\subseteq A\)✓\(\displaystyle 1\in A\)
The number 1 is an object. The set {1} can be a subsetDifference reversed
✕\(\displaystyle A\setminus B=B\setminus A\)✓\(\displaystyle \begin{gathered}A\setminus B=\{-2\}\\B\setminus A=\{3\}\end{gathered}\)
For A={-2,1,4} and B={1,3,4}, the order selects different elementsZero mistaken for empty
✕\(\displaystyle \{0\}=\varnothing\)✓\(\displaystyle n(\{0\})=1\)
The number zero is one element; the empty set has noneDifficulty levels
introMembership, subset, union and intersection of finite sets
standardComplement within a finite universe
challengeOrdered finite set difference. Real-interval notation is outside this lesson
Questions about this topic
Does order matter inside a set?+
No. {1,4} and {4,1} describe the same set. Ascending order makes a numerical set easier to check
Does order matter in a difference?+
Yes. A difference B starts with A; B difference A starts with B.
Can a set be its own subset?+
Yes. The subset-or-equal symbol allows equality
Can the intersection be empty?+
Yes, when the two sets share no elements
Why must a complement name a universe?+
The universe determines which outside elements are allowed. Changing the universe can change the complement
Can I use a calculator?+
No. This topic is practised without a calculator
Which year level and chapter is this?+
Year 9 Mathematics, chapter 9: Probability and statistics
In this chapter
Finite events, probability assignments and complementsDescribe and order probability values9B Venn diagrams and tables: Venn diagrams, two-way tables and set notation for two eventsComplete and use a two-way probability tableCompound events, the addition rule and mutual exclusivity9C Set notation: Event notation and shaded regionsFinite sets and set operationsBuild two-step sample spaces with and without replacementUse two-step probability treesExperimental probability from observed frequenciesDesign and interpret a probability simulationFind the mean and the median, and choose between themWork backwards from a mean or medianMode, bimodal and no-mode data, and the effect of an outlierConstruct an ordered stem plot from raw dataConstruct and read back-to-back stem-and-leaf plotsDescribe the shape of a distributionGroup numerical data into class intervalsConstruct a histogram from a frequency tableRead values from a histogramFind the range, the quartiles and the IQRProduce a five-number summary and construct a boxplotRead values and percentages from a boxplotCompare a numerical variable across groupsSurveys and statistical investigations