Starter
Draw a 3-way Venn Diagram as shown below, for the categories indicated.
Place the numbers from 1-20 in the appropriate regions. Numbers not in
any group should be placed outside the circles (but inside the rectangle)
Square numbers
Even numbers
18
8
6
7
16
14
12
1
20
3
13
15
19
4
10
2
17
9
5
11
Factors of 20
Set Theory
• Set theory is a branch of mathematics that
focuses heavily on logic
• It is often combined with Venn diagrams and
therefore can be used with probability
• There is some new notation for you to learn
• Today we will be focusing on this notation and
deciding whether some statements are true or
false
Set Theory
• A set is a collection of items
 Let A = the set of days in the week that begin with the letter T
 In set notation: 𝐴 = 𝑇𝑢𝑒𝑠𝑑𝑎𝑦, 𝑇ℎ𝑢𝑟𝑠𝑑𝑎𝑦
 The items in the brackets are separated with commas and are
called the ‘members’ or ‘elements’ of the set
 The number of elements in the set is denoted by n(A)
 So for the example above, n(A) = 2
Set Theory
𝐴 = 𝑇𝑢𝑒𝑠𝑑𝑎𝑦, 𝑇ℎ𝑢𝑟𝑠𝑑𝑎𝑦
Set notation
General
Applied to the example
𝑛 𝐴
‘The number of
members in the
set’
𝑛 𝐴 =2
∈
‘is an element
(member) of’
𝑇𝑢𝑒𝑠𝑑𝑎𝑦 ∈ 𝐴
∉
‘is not an
element
(member) of’
𝐹𝑟𝑖𝑑𝑎𝑦 ∉ 𝐴
𝑛 𝐴 ′𝑡ℎ𝑒 𝑛𝑢𝑚𝑏𝑒𝑟 𝑜𝑓 𝑚𝑒𝑚𝑏𝑒𝑟𝑠 𝑜𝑓 𝑎 𝑠𝑒𝑡′
∈ ′𝑖𝑠 𝑎 𝑚𝑒𝑚𝑏𝑒𝑟 𝑜𝑓′
∉
′
𝑖𝑠 𝑛𝑜𝑡 𝑎 𝑚𝑒𝑚𝑏𝑒𝑟 𝑜𝑓′
∅
This letter is
Norwegian, but
we just call it a
‘slashed zero’
This is the
Greek lowercase
letter ‘Xi’
ξ
Set Theory
This represents an
empty set
𝐼𝑓: 𝐴 = 𝐷𝑎𝑦𝑠 𝑜𝑓 𝑡ℎ𝑒 𝑤𝑒𝑒𝑘 𝑏𝑒𝑔𝑖𝑛𝑛𝑖𝑛𝑔 𝑤𝑖𝑡ℎ 𝑎 𝑃
𝑇ℎ𝑒𝑛: 𝐴 = ∅
This represents the
universal set –
everything being
considered in a
problem
 This is usually used
with a Venn diagram
𝐼𝑓: ξ = 𝐷𝑎𝑦𝑠 𝑜𝑓 𝑡ℎ𝑒 𝑤𝑒𝑒𝑘
𝐴𝑛𝑑: 𝐴 = 𝑇𝑢𝑒𝑠𝑑𝑎𝑦, 𝑇ℎ𝑢𝑟𝑠𝑑𝑎𝑦
ξ
Monday
Saturday
Friday
A
Thursday
Tuesday
Notice the
symbols used
Sunday
Wednesday
𝑛 𝐴 ′𝑡ℎ𝑒 𝑛𝑢𝑚𝑏𝑒𝑟 𝑜𝑓 𝑚𝑒𝑚𝑏𝑒𝑟𝑠 𝑜𝑓 𝑎 𝑠𝑒𝑡′
∈ ′𝑖𝑠 𝑎 𝑚𝑒𝑚𝑏𝑒𝑟 𝑜𝑓′
∉
′
𝑖𝑠 𝑛𝑜𝑡 𝑎 𝑚𝑒𝑚𝑏𝑒𝑟 𝑜𝑓′
ξ 𝑟𝑒𝑝𝑟𝑒𝑠𝑒𝑛𝑡𝑠 𝑡ℎ𝑒 𝑢𝑛𝑖𝑣𝑒𝑟𝑠𝑎𝑙 𝑠𝑒𝑡
∅ 𝑟𝑒𝑝𝑟𝑒𝑠𝑒𝑛𝑡𝑠 𝑎𝑛 𝑒𝑚𝑝𝑡𝑦 𝑠𝑒𝑡
Set Theory
List this set: ‘Factors of 18’
{1, 2, 3, 6, 9, 18}
Describe this set: {2, 3, 5, 7, 11, …} ‘Prime numbers’
Describe this set: {2, 3, 5, 7, 11}
‘Prime numbers less than 12 (or 13)’
True or False?
𝑻𝒓𝒊𝒂𝒏𝒈𝒍𝒆 ∈ {𝑸𝒖𝒂𝒅𝒓𝒊𝒍𝒂𝒕𝒆𝒓𝒂𝒍𝒔} False – it is not a member
True or False?
𝒙 ∈ {𝑭𝒂𝒄𝒕𝒐𝒓𝒔 𝒐𝒇 𝒙𝟐 − 𝟓𝒙} True – x is a factor of both terms
True or False?
𝑫𝒂𝒚𝒔 𝒐𝒇 𝒕𝒉𝒆 𝒘𝒆𝒆𝒌 𝒘𝒊𝒕𝒉 𝟓 𝒍𝒆𝒕𝒕𝒆𝒓𝒔 = ∅
True – the set has
no members
(in English)
Plenary
Try to show the following sets on a single Venn Diagram. The
complete set is the numbers from 1-20. Describe the empty set
that is on your diagram.
𝐴 = {𝐹𝑎𝑐𝑡𝑜𝑟𝑠 𝑜𝑓 14}
ξ
𝐵 = {𝑃𝑟𝑖𝑚𝑒 𝑛𝑢𝑚𝑏𝑒𝑟𝑠}
A
4
7
1
8
15
20
B
2
14
6
𝐶 = {𝑂𝑑𝑑 𝑛𝑢𝑚𝑏𝑒𝑟𝑠}
16
3
17 11
5
19 13
9
12
18
10
C
𝑃𝑟𝑖𝑚𝑒, 𝑏𝑢𝑡 𝑛𝑜𝑡 𝑎 𝐹𝑎𝑐𝑡𝑜𝑟 𝑜𝑓 14 𝑜𝑟 𝑂𝑑𝑑 = ∅
Plenary
No, this is not dingbats!
This is Gödel's
ontological proof of
god’s existence!
(Ontological means to
do with the theory of
existence)
Summary
• We have learnt about Set Theory
• We have seen some of the notation we
will be using
• We have also looked at Venn diagrams
which will be crucial over the next few
lessons