1.5 Set language and notation

Learning objectives

1.5A The definition of a setunderstand the definition of a set 1.5B Set notationuse the set notation ∪, ∩, ∈ and ∉, with E for the universal set and ∅ for the empty set 1.5C Universal and empty setsunderstand the concept of the universal set and the empty set and the symbols for these sets 1.5D Complements of setsunderstand and use the complement of a set, using the notation A'1.5E Venn diagrams to represent setsuse Venn diagrams to represent sets

A set is a well-defined collection of distinct objects, called elements or members. Curly brackets list the elements, for example $P=\{2,3,5,7\}$ .

Description	Set
vowels in the word MATHEMATICS	$\{A,E,I\}$
positive factors of $12$	$\{1,2,3,4,6,12\}$

A rule must decide unambiguously whether an object belongs. ‘Prime numbers below 10’ defines a set; ‘nice numbers’ does not unless ‘nice’ is precisely defined.

Order and repetition do not change a set: $\{1,2,3\}=\{3,2,1,1\}$ . Each distinct element is listed once in standard form.

The braces name the collection, not a calculation. $\{2,4\}$ is a set with two elements, whereas $2+4$ is a numerical expression.

Set symbols state membership and combine collections precisely.

Notation	Meaning
$x\in A$ / $x\notin A$	$x$ is / is not an element of $A$
$A\cap B$	elements in both $A$ and $B$
$A\cup B$	elements in $A$ or $B$ , including both
$E$	the universal set for the problem
$\varnothing$	the empty set, with no elements

If $A=\{1,2,3,4\}$ and $B=\{3,4,5\}$ , then $A\cap B=\{3,4\}$ and $A\cup B=\{1,2,3,4,5\}$ .

Membership relates an object to a set: $3\in A$ . Intersection and union relate two sets: $A\cap B$ and $A\cup B$ .

In set language, ‘or’ is inclusive: an element in both sets belongs to $A\cup B$ . Do not omit the overlap.

The universal set $E$ contains every element under consideration in a particular problem. Its contents depend on the stated context.

The empty set $\varnothing$ contains no elements, so $n(\varnothing)=0$ . It can arise when two sets share no members: $A\cap B=\varnothing$ .

Situation	Result
$E=\{1,2,3,4,5,6\}$ , $A=$ even numbers	$A=\{2,4,6\}$
$A=\{2,4,6\}$ , $B=\{1,3,5\}$	$A\cap B=\varnothing$

Every set used in the problem is interpreted inside $E$ . Changing $E$ can change what lies outside a named set.

$\varnothing$ is not the same as $\{0\}$ . The first has zero elements; the second has one element, namely zero.

The complement $A'$ is the set of all elements in the universal set $E$ that are not in $A$ .

Given	Complement
$E=\{1,2,3,4,5,6,7,8\}$
$A=\{2,4,6,8\}$	$A'=\{1,3,5,7\}$

On a Venn diagram, shade every region inside the universal rectangle but outside the circle for $A$ to represent $A'$ .

Read operations from the inside out. $(A\cup B)'$ means everything outside both circles, while $A'\cap B$ means in $B$ but not in $A$ .

A complement is relative to $E$ , not an unlimited collection of every object that is not in $A$ .

A Venn diagram represents sets as regions inside the universal rectangle. Overlapping circles show elements that satisfy more than one set condition.

Region	Meaning
overlap of $A$ and $B$	$A\cap B$
anywhere in either circle	$A\cup B$
in $A$ but outside $B$	$A\cap B'$
outside both circles	$(A\cup B)'$

Place intersection elements first, then elements belonging to only one set, then elements outside all named sets. Check every element of $E$ appears exactly once.

With three sets, begin at the central triple intersection, then fill pair-only regions, single-set regions and finally the outside region.

An element in an overlap is not copied into each circle’s separate region. Its one position already shows membership of both sets.

An algebraic set is defined by a condition on its elements, for example $A=\{x:x\in\mathbb{Z},\ 1\le x<6\}=\{1,2,3,4,5\}$ .

If every element of $A$ is also an element of $B$ , then $A$ is a subset of $B$ , written $A\subset B$ in this specification.

Sets	Decision
$A=\{2,4,6\}$ , $B=\{1,2,3,4,5,6\}$	$A\subset B$
$C=\{2,7\}$ , same $B$	$C\not\subset B$ because $7\notin B$

Translate the algebraic rule into allowed values, respecting the stated number domain. To disprove a subset claim, one counterexample is enough.

Do not confuse membership with subset notation: $2\in A$ , but $\{2\}\subset A$ .

A number written in a Venn region counts elements in that exact region. Totals for a set are found by adding every region inside its circle.

Information	Region value
$n(A)=28$ , $n(A\cap B)=9$	$A$ only $=28-9=19$
$n(B)=21$ , $n(A\cap B)=9$	$B$ only $=21-9=12$
$n(E)=40$	outside $=40-(19+9+12)=0$

Work from the most overlapped region outward. Subtract known overlap counts from set totals, then subtract all circle regions from the universal total.

For three sets, fill the triple intersection first. A stated pair intersection includes the triple region, so subtract it to obtain the pair-only region.

Adding $n(A)+n(B)$ double-counts the overlap. For two sets, $n(A\cup B)=n(A)+n(B)-n(A\cap B)$ .

$n(A)$ means the number of distinct elements in set $A$ ; it is a number, not a set.

Set expression	Count
$A=\{2,3,5,7\}$	$n(A)=4$
$A\cap B=\{3,5\}$	$n(A\cap B)=2$
$A\cap B=\varnothing$	$n(A\cap B)=0$

On a Venn diagram, add the counts in every region described by the expression inside $n(\ )$ . For $n(A')$ , add all regions outside $A$ but still inside $E$ .

Count distinct members only. Repeated writing does not create extra elements in a set.

$n(A)$ is not the same object as $A$ : if $A=\{4,6,8\}$ , then $A$ is a set and $n(A)=3$ .

In a practical problem, each set represents a precisely defined property, such as students studying French or customers buying a product.

Words	Set expression
both football and tennis	$F\cap T$
football or tennis or both	$F\cup T$
tennis but not football	$T\cap F'$
neither activity	$(F\cup T)'$

Define the universal group and each set before calculating. Translate one phrase at a time, place or count the overlap first, and check that all regions sum to the stated total.

Return to the context when stating an answer: a region count should be described as people, items or outcomes, with any stated units.

Everyday ‘or’ can sound exclusive, but set union is inclusive unless the problem explicitly says ‘but not both’ or ‘exactly one’.

Next Topic

—1.6 PercentagesRead

1.5 Set language and notation

Learning objectives

Define and describe a set

Read and write set notation

Use universal and empty sets

Find complements of sets

Place elements in Venn diagrams

Define algebraic sets and subsets

Calculate region counts in Venn diagrams

Interpret the notation n(A)

Model practical situations with sets

Next Topic