Sets: ∈ or ⊆, number sets, ∩ and ∪

Understand it

A set is a collection of distinct objects, written between braces: $A=\{1,2,3\}$. Order does not matter and repeats do not count, so $\{1,2,2,3\}$ is the same set as $\{3,2,1\}$. The objects are the elements.

Two different relations, and the exam tests the difference every time:

• $\in$ links an element to a set: $2\in A$ reads “2 is an element of $A$”. Its negation is $\notin$.
• $\subseteq$ links a set to a set: $B\subseteq A$ means every element of $B$ is also in $A$.

Read the left side first. A bare element (a number) takes $\in$ or $\notin$. Anything in braces takes $\subseteq$. The empty set $\varnothing$ has no elements, and it is a subset of every set, so $\varnothing\subseteq A$ is always true while $\varnothing\in A$ is false unless $\varnothing$ is itself listed in $A$.

Sets are often described by a rule, called set-builder notation. $A=\{x\mid x^2-9=0\}$ means “all $x$ such that $x^2-9=0$”. Solve the rule first and list the elements: $A=\{-3,3\}$. Only then compare.

Number sets. $\mathbb{N}$ is the natural numbers $0,1,2,\dots$ (zero is included); $\mathbb{N}^*$ (also written $\mathbb{N}_+$) is $1,2,3,\dots$ without zero; $\mathbb{Z}$ is all integers; $\mathbb{Q}$ is every fraction of two integers with a nonzero denominator; $\mathbb{R}$ is all real numbers, which also include irrational numbers such as $\sqrt2$ and $\pi$. Each set sits inside the next, so $\mathbb{N}\subseteq\mathbb{Z}\subseteq\mathbb{Q}\subseteq\mathbb{R}$.

Counting subsets. Each element is either in a subset or out of it, two choices per element, so a set with $n$ elements has $2^n$ subsets, counting $\varnothing$ and the set itself. Leave out the set itself and $2^n-1$ proper subsets remain; leave out $\varnothing$ as well and $2^n-2$ non-empty proper subsets remain.

Intersection and union. $A\cap B$ keeps only what is in both sets (“and”). $A\cup B$ collects everything that is in either set (“or”), writing shared elements once.

Intervals. Inequalities are written as intervals. A round bracket leaves the endpoint out and a square bracket keeps it: $[a,b)$ means $a\le x<b$. Infinity always gets a round bracket. On a number line, an open circle means “not included” and a filled circle means “included”. Draw both sets on one line, shade where they overlap for $\cap$, and shade everything covered for $\cup$.

Essential formulas

See it

Worked examples

Try it yourself

Work it on paper first. Open a hint only when you are stuck, then compare with the solution.

Let $M=\{x\in\mathbb{N}\mid x<3\}$ and $N=[1,4)$. List $M$, then find $M\cap N$ and $M\cup N$.

In short

• $\in$ is for elements, $\subseteq$ is for sets; $\varnothing\subseteq$ every set.
• $\mathbb{N}$ includes $0$; $\mathbb{N}^*$ does not.
• An $n$-element set has $2^n$ subsets, $2^n-1$ proper and $2^n-2$ non-empty proper.
• $A\cap B$ is the overlap (and); $A\cup B$ is everything covered (or).
• Square bracket includes an endpoint, round bracket excludes it; infinity is always round.

Knowledge check

4 questions on this lesson, marked as you go. Take them in ACE CSCA with a free account: passing them completes the lesson, and your progress is saved on every device.