Set notation, operations and Venn diagrams

Sets and their notation

A set is a collection of objects, called elements (or members). The elements go inside curly brackets, separated by commas.

Order does not matter and an element is not listed twice. Sets are usually named with capital letters.

The universal set $U$ is the set of everything we are talking about in a problem. Every other set in the problem lives inside $U$.

A set can also be described by a rule. For example, $B = \{x \in \mathbb{N} : 1 \le x \le 6\}$ means "$x$ is a natural number with $1 \le x \le 6$", so $B = \{1, 2, 3, 4, 5, 6\}$. Read the colon as "such that".

Union, intersection, difference and complement

Four operations combine sets. Take $U = \{1, 2, \dots, 12\}$, $A = \{2, 3, 5, 7, 11\}$ (the primes) and $B = \{3, 6, 9, 12\}$ (the multiples of 3).

Why does $A \cup B$ list 3 only once? Because a set never repeats an element.

(Answer to the last step: $A \setminus B = \{2, 5, 7, 11\}$ and $B \setminus A = \{6, 9, 12\}$. They are different, so the order matters.)

Venn diagrams with two sets

A Venn diagram draws $U$ as a rectangle and each set as a circle. Where circles overlap we have the intersection. Two overlapping circles split $U$ into four regions.

To fill in a diagram from numbers, start in the middle (the intersection) and work outwards:

1. Put the number in $A \cap B$ in the overlap.

2. Subtract it from $\#A$ to get "A only", and from $\#B$ to get "B only".

3. Put "neither" in $U$ outside both circles, found by subtracting the others from $\#U$.

This works because every element of $U$ lies in exactly one region, so the four numbers add up to $\#U$.

For the union, we count the overlap twice if we simply add, so we take it off once:

(Answer: $\frac{10}{40} = \frac{1}{4}$.) In general, for a Venn diagram with numbers,

Venn diagrams with three sets

Three overlapping circles give eight regions: three "only" regions, three regions where exactly two circles overlap, the centre where all three overlap, and the outside.

Always fill in the centre $A \cap B \cap C$ first. Then do the regions where exactly two sets overlap, subtracting the centre. Then the "only" regions. Finally find the outside from $\#U$.

(Answer: $5 + 4 + 3 = 12$.)

When a three-set problem gives a rule for each set (for example "multiples of 3"), list the elements of each set first. Then place each element of $U$ one at a time, starting with those that belong to the most sets, and cross each one off as you go. Every element of $U$ must appear exactly once.

Problems with unknowns

Sometimes a region contains an unknown. Write each region as an expression in $x$. The regions must add up to $\#U$, which gives an equation to solve.

(Answer: $\#(A \cup B) = 35$, and $P = \frac{20}{45} = \frac{4}{9}$.)

Complements of unions and intersections

Two results connect complements with $\cup$ and $\cap$. They are true for all sets $A$ and $B$:

Why? Number the four regions of the Venn diagram.

• $A \cup B$ is regions 1, 2, 3, so $(A \cup B)'$ is region 4. And $A'$ is regions 3, 4 while $B'$ is regions 1, 4, so $A' \cap B'$ is also region 4. ✓

• $A \cap B$ is region 2, so $(A \cap B)'$ is regions 1, 3, 4. $A' \cup B'$ is regions 3, 4 together with 1, 4, which is again regions 1, 3, 4. ✓

Summary

• A set is written $\{\dots\}$. Use $\in$ for an element, $\subset$ for a subset and $\#A$ for the number of elements.

• $A \cup B$ is "or", $A \cap B$ is "and", $A \setminus B$ is "in $A$ not $B$", $A'$ is "not in $A$".

• Fill Venn diagrams from the middle outwards, and check that all regions add up to $\#U$.

• $\#(A \cup B) = \#A + \#B - \#(A \cap B)$. With three sets, take care to subtract the centre.

• Probability from a Venn diagram: number in the region $\div\ \#U$.

• $(A \cup B)' = A' \cap B'$ and $(A \cap B)' = A' \cup B'$. An example verifies, but only an argument for all sets proves.

Back to the start: with 20 in the choir and 15 in the band out of 30, at least $20 + 15 - 30 = 5$ students must be in both.