Sets Class 11: notes and important questions

Sets: revision notes

1. Sets and their representations

A set is a well-defined collection of objects (its elements); $a\in A$ means $a$ is an element of $A$. Roster form lists the elements once each, in any order: $\{2,3,5,7\}$. Set-builder form states the property: $\{x:x\text{ is a prime},x<10\}$.

2. Kinds of sets

• Empty set $\varphi =\{\}$: no elements (for example $\{x\in \mathbb{R}:x^2=-1\}$).
• Finite set: a definite number of elements $n(A)$; otherwise infinite ($\mathbb{N},\mathbb{Z},\mathbb{Q},\mathbb{R}$).
• Equal sets: exactly the same elements. $\{1,2,2,3\}=\{3,1,2\}$.

3. Subsets and intervals

$A\subset B$ if every element of $A$ is in $B$; $A=B$ exactly when $A\subset B$ and $B\subset A$. For every set $A$, both $\varphi$ and $A$ itself are subsets of $A$. Intervals are subsets of $\mathbb{R}$:

• $(a,b)=\{x:a<x<b\}$, $[a,b]=\{x:a\le x\le b\}$, $[a,b)=\{x:a\le x<b\}$, $(a,b]=\{x:a<x\le b\}$; a round bracket leaves the end-point out.

4. Operations and Venn diagrams

• Union $A\cup B=\{x:x\in A\text{ or }x\in B\}$; intersection $A\cap B=\{x:x\in A\text{ and }x\in B\}$; $A$ and $B$ are disjoint if $A\cap B=\varphi$.
• Difference $A-B=\{x:x\in A,x\notin B\}$.
• Laws: commutative, associative, $A\cup \varphi =A$, $A\cap U=A$, distributive: $A\cap (B\cup C)=(A\cap B)\cup (A\cap C)$ and $A\cup (B\cap C)=(A\cup B)\cap (A\cup C)$.
• A Venn diagram shows $U$ as a rectangle and sets as circles; shade the region asked for.

5. Complement

$A^{\prime }=U-A=\{x\in U:x\notin A\}$. Properties: $A\cup A^{\prime }=U$, $A\cap A^{\prime }=\varphi$, $(A^{\prime }{)}^{\prime }=A$, ${\varphi }^{\prime }=U$, $U^{\prime }=\varphi$, and De Morgan's laws $(A\cup B{)}^{\prime }=A^{\prime }\cap B^{\prime }$, $(A\cap B{)}^{\prime }=A^{\prime }\cup B^{\prime }$. Also $A-B=A\cap B^{\prime }$.

Worked example 1

Write $\{x:x\in \mathbb{Z},x^2<10\}$ in roster form.

$x^2<10$ for $x=-3,-2,\dots ,3$: $\{-3,-2,-1,0,1,2,3\}$.

Worked example 2

Find $A\cap B$ and $A-B$ for $A=[0,4)$, $B=(2,6]$.

$A\cap B=(2,4)$, $A-B=[0,2]$.

Worked example 3

With $U=\{1,\dots ,8\}$, $A=\{1,2,3\}$, $B=\{3,4\}$, find $(A\cup B{)}^{\prime }$.

$A\cup B=\{1,2,3,4\}$, so $(A\cup B{)}^{\prime }=\{5,6,7,8\}$.

Common errors

• Repeating elements in roster form, or treating $\{\varphi \}$ as empty (it has one element).
• Getting the end-points of intervals wrong in intersections and differences.
• Writing $(A\cup B{)}^{\prime }=A^{\prime }\cup B^{\prime }$: De Morgan's laws swap union and intersection.
• Confusing $\in$ (element) with $\subset$ (subset).

Exam tips

• For interval questions, draw both intervals on one number line first.
• To prove a set identity, show "$x$ in the left side $\Rightarrow x$ in the right side" and the reverse.