Sets — Mathslive.ie

Section 1 of 4

Set Notation

A set is a collection of elements. Learn the symbols — every question uses them.

The symbols

$\cap$ intersection (in both) $\cup$ union (in either) $A'$ complement (outside $A$) $\#A$ cardinal (how many) $A\setminus B$ difference (in $A$, not $B$) $\subset$ subset $\varnothing$ the null set $\{\ \}$.

Section 2 of 4

Listing Elements

Take $A = \{1,2,3,4\}$, $B = \{2,3,5\}$, $C = \{1,3,4,5,6\}$.

Worked example — difference and intersection

List $(A\setminus B)\cup(C\cap B)$.

$A\setminus B = \{1,4\}$ (in $A$ but not $B$)

$C\cap B = \{3,5\}$ (in both)

Union them: $\{1,4\}\cup\{3,5\}$

$\{1,3,4,5\}$

Worked example — union and difference

List $(A\cup B)\cap(C\setminus B)$.

$A\cup B = \{1,2,3,4,5\}$

$C\setminus B = \{1,4,6\}$

Intersection (in both): $\{1,2,3,4,5\}\cap\{1,4,6\}$

$\{1,4\}$

Section 3 of 4

Venn Diagrams & Shading

A Venn diagram shows two sets inside the universal set $U$. Shade the region the question asks for.

Worked example — shade the complement of a union

Shade $(P\cup Q)'$ — everything outside both $P$ and $Q$.

$P\cup Q$ is everything inside either circle

The dash $'$ means the complement — so shade everything outside both circles

The shaded region is outside both circles, inside $U$.

Section 4 of 4

Venn Diagrams with Numbers

Put an expression in each region, add them to the total, and solve.

Now you try

You try

$\#U=20$, $\#(P\cap Q)=x$, $\#(P\setminus Q)=2x$, $\#((P\cup Q)')=4$, and $\#Q = 2(\#P)$. Find $\#Q$.

Pen and paper out — try it before you reveal.

$\#P = 3x$, so $\#Q = 6x$, giving the right region $5x$

All regions add to $\#U$: $2x + x + 5x + 4 = 20$

$8x = 16 \Rightarrow x = 2$

$\#Q = 6x = 6(2)$

$\#Q = 12$

You try

A leisure centre has $110$ members. The weights room $W$ is used by $82$, the pool $S$ by $57$, and $15$ use neither. How many use both?

Pen and paper out — try it before you reveal.

Let $x$ use both. The four regions add to $110$:

$(82-x) + x + (57-x) + 15 = 110$

$154 - x = 110 \Rightarrow x = 44$

$44$ use both facilities

That’s Sets.

The notation, listing elements, shading Venn diagrams, and region problems with numbers — the whole of Junior Cert sets.