Sets and Venn diagramsIB MYP Maths Extended: Revision notes

A set is a collection of objects called elements, written in curly brackets, for example $A=\{2,4,6,8\}$. The universal set $\xi$ contains every element being considered.

• $x\in A$: $x$ is an element of $A$
• $n(A)$: the number of elements in $A$
• $\varnothing$ or $\{\ \}$: the empty set Example: with $\xi=\{1,\dots,12\}$ and $A$ the even numbers, $A=\{2,4,6,8,10,12\}$ and $n(A)=6$.

• Union $A\cup B$: elements in $A$ or $B$ (or both).
• Intersection $A\cap B$: elements in both $A$ and $B$.
• Complement $A'$: elements of $\xi$ that are not in $A$. Example: $A=\{2,4,6,8,10,12\}$ and $B=\{3,6,9,12\}$. $A\cap B=\{6,12\}$, $A\cup B=\{2,3,4,6,8,9,10,12\}$ and $(A\cup B)'=\{1,5,7,11\}$. The counting rule is $n(A\cup B)=n(A)+n(B)-n(A\cap B)$, because the intersection is counted twice if you just add.

A Venn diagram shows sets as overlapping circles inside a rectangle that represents $\xi$. Fill it in from the middle outwards:

1. Write the number in the overlap $A\cap B$ first.
2. Subtract it from each set total to find 'only $A$' and 'only $B$'.
3. Subtract the total of all regions from $n(\xi)$ to find the number outside the circles. Example: 44 students, $n(F)=22$, $n(B)=18$, 8 play both. Football only $=14$, basketball only $=10$, both $=8$, neither $=44-32=12$.

With three sets, start with the region where all three overlap. Then work out each region where exactly two sets overlap by subtracting the all-three number from each pair total. Example: $n(H\cap T)=8$ includes the 5 in all three sets, so only $H$ and $T$ is $8-5=3$. Next find each 'only' region, for example $H$ only $=22-3-4-5=10$. Finally, find the number outside all circles by subtracting the total of the regions from $n(\xi)$.

Probabilities come from the numbers in the regions: $P(\text{event})=\frac{\text{number of elements in the event}}{n(\xi)}.$ Example: in the group of 44, $P(\text{exactly one sport})=\frac{14+10}{44}=\frac{6}{11}$. Use set notation to describe the event: 'art but not music' is $A\cap M'$, and 'neither' is $(M\cup A)'$. Add only the regions that belong to the event, and check that the regions do not overlap.