## What you need to know

### Set Notation

In maths, a set is a collection of things. The kind of sets you’ll need to be familiar with are sets of numbers. Students sometimes called set notation Venn diagram symbols as these are the symbols you need to know to succeed with Venn diagrams.

We express a set using ‘curly brackets’, i.e. if we wanted to write a set that contained the numbers 5, 13, 2, and 110, we would write

\{5, 13, 2, 110\}

A nice thing about sets is that order doesn’t matter. You could write those numbers in reverse order and the set would still be the same. Understanding how set notation works is really important when looking at Venn diagrams.

### Understanding Set Notation

We usually name sets using capital letters. To call the above set A, we would simply write

A=\{5, 13, 2, 110\}

Suppose we have two different sets, A and B. Then, the important set notation is as follows.

•A \cap B means ‘A and B’. It’s a new set that contains only the elements that are both in A and in B.

•A \cup B means ‘A or B’. It’s a new set that contains any element that appears anywhere in either A or B.

•A’ means ‘not in A’. It’s the set of all the elements that don’t appear in A.

You may be thinking that a set that contains elements not in A is going to be, well, a really big set. After all, there are a lot of the numbers that aren’t 5, 13, 2 or 110. However, we usually narrow down the numbers we consider with 1 more bit of notation.

•\xi is the universal set – it contains all the numbers we’re interested in at that time. In this context, A’ would only contain numbers from the universal set that aren’t in A (as opposed to every number ever that isn’t in A).

All of these things start to make a bit more sense when we see some examples.

### Set Notation for Inequalities

The other time set notation appears is in working with inequalities. The good news is, it’s pretty straightforward! To write the inequality x \leq 5 in set notation, we write

\{x : x\leq 5\}

So, you have to add 3 things: curly brackets, the variable, and a colon.

## Example 1: Set Notation

Let A=\{1, 2, 3, 8, 10, 12\} and B=\{6, 5, 4, 3, 2, 10\}. Write down the set

a)A\cap B,

b)A\cup B.

a) To recall, A\cap B contains every element that appears in both A and B. Looking at the sets, we see that there are three of them: 3, 2, and 10. Therefore, we have

A\cap B=\{2, 3, 10\}

b) A\cup B contains all the elements that are in either A or B. Writing the elements in – and it helps to cross them off with a pencil as you go – we get

A \cup B=\{1, 2, 3, 4, 5, 6, 8, 10, 12\}

**Note:** when a number appears in both sets, we don’t need to write it in twice. The numbers 2, 3 and 10 appear in both but we only count them once when working out A \cup B.

## Example 2: Sets Notation and Venn Diagrams

Below is a Venn diagram. State which numbers are in the sets:

a) P \cap Q

b) Q’

c) P\cup Q.

a) P\cap Q means all elements that are in both P and Q. Looking at the Venn diagram, this must be where the circles intersect. So, we get

P\cap Q=\{4, 5\}

b) Q’ means all numbers not in Q. All the numbers inside the Q circle are: 10, 6, 14, 4, and 5 (the ones in the intersection count). The numbers not in there are then

Q’=\{2, 8, 7, 1\}

We only include numbers in the universal set, \xi, as seen on the top-left.

c) P\cup Q means any numbers that appear in either P or Q. So, anything in either circle. We get

P\cup Q=\{2, 8, 4, 5, 10, 6, 14\}

## Example 3: Sets and Inequalities

Solve the inequality 3x-7>11. Write your answer using set notation.

We solve linear inequalities like we do linear equations. Adding 7 to both sides, we get

3x>7+11=18

Then, dividing both sides by 3, we get

x>6

Now, to express this in set notation, we want to put “x :” before it and wrap the whole thing in curly brackets:

\{x : x>6\}

### Example Questions

1) \xi = \{103, 104, 105, 109, 110, 112, 114\}

A = \text{ even numbers}

B = \{103, 112, 114\}

State which numbers belong to the following:

a) A\cup B

b) A’

a) The first thing we need to do is work out which numbers belong to the subset A. Since A is the subset consisting of even numbers, then the following numbers belong in subset A:

A=\{104, 110, 112, 114\}

The \cup symbol tells us that we need to combine subsets A and B, so when we combine the even numbers from subset A with the 3 numbers in subset B, we have the following numbers:

A\cup B=\{103, 104, 110, 112, 114\}

(Note that you don’t write the numbers 112 and 114 twice.)

b) The dash after the A means that we are interested in the set of numbers that are not in subset A.

Since we already know that A is the subset of even numbers, A' is the group of odd numbers from the universal set \xi:

A’=\{103, 105, 109\}

2) \xi = \{1, 2, 3, 5, 7, 8, 9, 10, 12\}

V = \text{ prime numbers}

V\cap W = \{3, 5, 7\}

V\cup W = \{1, 2, 3, 5, 7, 10, 12\}

Draw a Venn diagram to display this information.

The first thing we need to do is work out which numbers from the universal set are in subset V, the set of prime numbers:

V = \{2, 3, 5, 7\}

(This part may be challenging if you do not know what a prime number is. Remember that a prime number is a number which is only divisible by itself and 1. The first prime number is 2; 1 is not a prime number.)

Our Venn diagram should consist of 2 overlapping circles, one labelled ‘V’ and the other labelled ‘W’, with a rectangle around them both. The rectangle should be labelled with the Greek letter \xi, the symbol of the universal set.

We are told in the question that V\cap W=\{3, 5, 7\}. This means 3, 5 and 7 are in both circles. This means that these numbers must be placed in the small area where the two circles overlap.

This means that the only remaining number from the subset V, the number 2, still needs to be placed. The number 2 needs to be placed inside the V circle, but outside the W circle (in other words, not in the intersection).

Now that we have organised all the elements of subset V, all the other numbers that are part of V\cup W need to be placed inside W, but outside V (again, not in the intersection). These values that we need to be place here are the numbers 1, 10, and 12.

Finally, there are still some numbers in the universal set \xi which have not yet been placed. All these numbers, the numbers 8 and 9, need to placed outside the circles, but still inside the rectangle.

Your completed Venn diagram should be similar to the below:

3) Write the following inequalities using set notation.

a) x \geq 12

b) z < -2

c) a > 0

d) 13 < x

The process for all 4 questions will be the same. Write the variable followed by a colon before the inequality, and then put everything inside curly brackets { }. The results are as follows:

a) \{x : x \geq 12\}

b) \{z : z < -2\}

c) \{a : a > 0\}

d) \{x : 13 < x\}

4) Draw a Venn diagram to show the following sets where \xi = {x : x is an integer, 1 < x < 21}

A = {x : x is a prime number}

B = {x : x is a factor of 24}

C = {x : x is a square number}

The question may appear a little bit off-putting due to the set notation. The key facts we need to understand are that:

- the universal set (all the numbers in the set) must be greater than 1 (so from 2 onwards), but less than 21 (so up to and including 20).

- all the numbers in set A are prime numbers.

- all the numbers in set B are factors of 24 (numbers that 24 can be divided by).

- all the numbers in set C are square numbers.

It would probably be useful at this stage to write down which numbers fit into each set, and then see which numbers appear in multiple sets.

The prime numbers from 2 – 20 are: 2, 3, 5, 7, 11, 13, 17 and 19. These are the numbers that will appear in set A.

The factors of 24 are: 2, 3, 4, 6, 8 and 12. These are the numbers that will appear in set B.

The square numbers are: 4, 9 and 16. These are the numbers that will appear in set C.

The first thing to check is to see if there are any numbers that belong to all three sets. There aren’t any, so the intersection of the three circles will be empty.

Having done this, look for any numbers that are common to 2 sets. Sets A and B share numbers 2 and 3, and sets B and C share the number 4. Therefore, we need to write 2 and 3 in the intersection of circle A and circle B, and the number 4 in the intersection of circle B and circle C.

In set A, we have already input numbers 2 and 3 on the Venn diagram, so we now need to input numbers 5, 7, 11, 13, 17 and 19. These numbers should be placed in the A circle, but not in any of the intersections.

In set B, we have already input numbers 2, 3 and 4 on the Venn diagram, so we now need to input numbers 6, 8 and 12. These numbers should be placed in the B circle, but not in any of the intersections.

In set C, we have input the number 4 on the Venn diagram, so we now need to input numbers 9 and 16. These numbers should be placed in the C circle, but not in any of the intersections.

The completed Venn diagram should look like the below:

5) Shade the region that represents (A'\cap B').

To work out the what area to shade, we need to be really clear on what (A'\cap B') means:

A' means everything ‘not in A‘.

B' means everything ‘not in B‘.

\cap means ‘the intersection of’.

Therefore, we need to shade anything that is not in A and which is also not in B. Anything not in A and not in B is everything outside of the circles, so the Venn diagram should be shaded as follows:

### Worksheets and Exam Questions

#### (NEW) Set Notation Exam Style Questions - MME

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