2 Sets

Learning objectives

2A SetsThe idea of a set 2B Set language and notationSet language and notation Questions may be set involving these ideas in the abstract or derived from practical situations 2C Union and intersectionUnion and intersection of sets Understand sets defined in algebraic terms 2D Number of elementsNumber of elements in a set Use the notation n(A)2E Complementary setsComplementary sets Use the notation A′2F SubsetsSubsets 2G Universal and null setsUniversal set, null set 2H Venn diagramsVenn diagrams and their use in simple logical problems 2I Set symbolsUse of symbols to represent sets

A set is a well-defined collection of distinct objects, called elements. 'Well-defined' means that membership can be decided unambiguously.

The vowels in the word MATHEMATICS form the set $\{A,E,I\}$ : repeated letters are written once. The collection 'interesting numbers' is not well-defined unless a precise rule for interesting is supplied.

The order used to list elements does not change the set, so $\{2,4,6\}=\{6,2,4\}$ . Braces describe the collection; they do not mean an ordered list.

Form	Example	Meaning
words	$A$ is the set of positive even integers below $10$	states the membership rule
roster	$A=\{2,4,6,8\}$	lists every element once
set-builder	$A=\{x:x ext{ is a positive even integer and }x<10\}$	gives a rule after the colon

In a practical situation, define what one element represents before forming the set. In an abstract set, test each candidate against every condition in the rule.

Dots such as $\{2,4,6,\ldots\}$ are safe only when the continuation rule is unambiguous. Do not confuse an element such as $2$ with the one-element set $\{2\}$ .

Operation	Membership test	Venn region
$A\cap B$	in both $A$ and $B$	overlap only
$A\cup B$	in $A$ or $B$ or both	every region inside either set

A={1,2,3,4},\quad B={3,4,5}\ A\cap B={3,4},\qquad A\cup B={1,2,3,4,5}

For algebraically defined sets, translate each rule into a membership condition. Intersection means satisfying both conditions simultaneously; union means satisfying at least one.

The mathematical word 'or' in a union is inclusive: elements in the overlap are included. List each element only once.

The notation $n(A)$ means the number of distinct elements in set $A$ , not the sum of their numerical values.

A={2,4,6,8}\quad\Longrightarrow\quad n(A)=4

n(A\cup B)=n(A)+n(B)-n(A\cap B)

The intersection is subtracted once because it was counted in both $n(A)$ and $n(B)$ . On a Venn diagram, add every disjoint region belonging to the required set exactly once.

An empty region contributes $0$ elements. A written zero inside a region is a count, not an element that must be counted as one.

The complement $A'$ contains every element of the universal set that is not in $A$ . Its meaning therefore depends on the chosen universal set.

\mathcal E={1,2,3,4,5,6},\quad A={2,4,6}\quad\Longrightarrow\quad A'={1,3,5}

Expression	Equivalent description
$(A\cup B)'$	outside both $A$ and $B$ ; $A'\cap B'$
$(A\cap B)'$	not in both together; $A'\cup B'$

Complement does not mean 'negative' or 'opposite'. It means outside the named set but still inside the universal set.

Set $A$ is a subset of set $B$ when every element of $A$ is also an element of $B$ . The empty set and the whole set are subsets of every set and of itself respectively.

ext{A set with }n ext{ elements has }2^n ext{ subsets.}

For $P=\{a,b\}$ , the complete subset list is $arnothing$ , $\{a\}$ , $\{b\}$ and $\{a,b\}$ . There are $2^2=4$ subsets.

Membership and subset are different: $a$ is an element of $P$ , while $\{a\}$ is a subset of $P$ . Reordering the same elements does not create another subset.

Set	Meaning	Key consequence
universal set $\mathcal E$	all elements currently under consideration	every named set is contained in it
empty set $arnothing$	a set with no elements	$n( arnothing)=0$

If $\mathcal E=\{1,2,3,4,5,6\}$ , $A$ is the even elements and $B$ is the odd elements, then $A\cap B= arnothing$ and $A\cup B=\mathcal E$ .

The empty set is not the same as $\{0\}$ . The first has no elements; the second has one element, the number zero. The universal set changes with the stated context.

A Venn diagram divides the universal set into mutually exclusive regions. For three sets, enter information from the most specific region outward.

Place the three-set intersection. 2. Fill each 'exactly two' region after subtracting the triple intersection when necessary. 3. Fill each single-set-only region. 4. Put elements in none of the sets outside all circles but inside the rectangle. 5. Check every region against the totals.

n(\mathcal E)= ext{sum of all disjoint regions, including the outside region}

A value in an overlap belongs to every circle that contains it, but it is counted once in the universal total. Row 199715 supplied a diagram but no answer and was excluded as representative evidence.

Symbol	Read as
$x\in A$ / $x
otin A $\|$ x $is / is not an element of$ A$
$A\subseteq B$	$A$ is a subset of $B$
$A\cup B$ / $A\cap B$	union / intersection
$A'$	complement of $A$
$arnothing$	empty set
$n(A)$	number of elements in $A$
$\mathcal E$	universal set

Read a compound expression from its grouping outward. For example, $n(B'\cap C)$ asks for the number of elements that are in $C$ and not in $B$ .

Symbols describe different relationships: $3\in A$ can be true while $3\subseteq A$ is ill-formed because $3$ is an element, not a set. Preserve brackets when taking a complement of a combined set.

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—3 AlgebraRead

2 Sets

Learning objectives

Understand what makes a set

Describe sets by listing or by a rule

Find unions and intersections

Count the elements of a set

Take the complement within the universal set

Recognise and list subsets

Use universal and empty sets

Translate information into a Venn diagram

Read and combine set symbols

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