E1.2 Sets

Learning objectives

E1.2.1Understand and use set language, notation and Venn diagrams to describe• Understand and use set language, notation and Venn diagrams to describe sets and represent relationships between sets. Venn diagrams are limited to two or three sets. The following set notation will be used: • n(A) Number of elements in set A • ∈ “… is an element of …” • ∉ “… is not an element of …” • A′ Complement of set A • ∅ The empty set • Universal set • A ⊆ B A is a subset of B • A ⊈ B A is not a subset of B • A ∪ B Union of A and B • A ∩ B Intersection of A and B. Example definition of sets: A = {x: x is a natural number} B = {(x, y): y = mx + c} C = {x: a ⩽ x ⩽ b} D = {a, b, c, … }.

A set is a collection of distinct elements. In a Venn diagram, the rectangle is the universal set $\mathscr E$ and each circle contains the elements of one set. Overlaps show elements that satisfy more than one set description; regions outside a circle show elements that do not belong to that set.

Notation	Meaning
$n(A)$	number of elements in $A$
$x\in A$ / $x\notin A$	$x$ is / is not an element of $A$
$A'$	complement: elements of $\mathscr E$ that are not in $A$
$\varnothing$	empty set: a set with no elements
$A\subseteq B$ / $A\nsubseteq B$	every element of $A$ is / is not also in $B$
$A\cup B$	union: in $A$ or $B$ , including the overlap
$A\cap B$	intersection: in both $A$ and $B$

Sets may be listed, such as $D=\{a,b,c,\ldots\}$ , or defined by a rule. Examples are $A=\{x:x\text{ is a natural number}\}$ , $C=\{x:a\le x\le b\}$ , and $B=\{(x,y):y=mx+c\}$ . The colon means ‘such that’; an ordered pair $(x,y)$ is one element of $B$ .

To translate notation into a Venn region:

Work inside brackets first, then apply any complement mark.
For an intersection, keep only regions that satisfy every condition.
For a union, include regions that satisfy at least one condition.
For a complement, switch to the part of the universal set outside the named region.
With three sets, identify the most specific overlap first; for example, $A\cap B'\cap C$ is inside $A$ and $C$ but outside $B$ .

Let $\mathscr E=\{1,2,3,4,5,6,7,8,9,10\}$ , $A=\{2,4,6,8,10\}$ and $B=\{3,6,9\}$ . Then $A\cap B=\{6\}$ , $A\cup B=\{2,3,4,6,8,9,10\}$ , and $A'=\{1,3,5,7,9\}$ . Also $A\cap B'=\{2,4,8,10\}$ , so $n(A\cap B')=4$ .

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

The intersection is subtracted because it was counted once in $n(A)$ and once in $n(B)$ . When filling a numerical Venn diagram, place the deepest intersection first, then pair-only regions, then single-set regions, and finally the region outside all sets. This prevents an overlap from being counted twice.

$x\in A$ says that $x$ is one element; $X\subseteq A$ says that every element of set $X$ lies in $A$ . The empty set $\varnothing$ has no elements, whereas $\{0\}$ contains one element. A complement is always relative to the stated universal set, and ‘or’ in a union includes the intersection unless the question explicitly excludes it.

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—E1.3 Powers and rootsRead