Set Notation

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:

$Acup B={103, 104, 110, 112, 114}$

(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 $xi$:

$A’={103, 105, 109}$

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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:

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

(Remember that a prime number is a number which is only divisible by itself and 1.)

We are told in the question that $Vcap W={3, 5, 7}$. 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 $Vcup W$ 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:

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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) ${x : 13 < x}$

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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:  $2, , 3, , 5, , 7, , 11, ,  13, , 17, , 19$.  These are the numbers that will appear in set $A$.

The factors of $24$ are:  $2, , 3, , 4, , 6, , 8, ,  12$.  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 $C$. We can write these sets as:

$begin{aligned} A&= {2, , 3, , 5, , 7, , 11, ,  13, , 17, , 19} B&= {2, , 3, , 4, , 6, , 8, ,  12} C &= {4, , 9, , 16}end{aligned}$

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 $5, , 7, , 11, , 13, , 17, , 19$.  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:

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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:

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