GCSE Sets
In maths, it’s often useful to represent a collection of items.
We use curly braces to indicate a set of items...
A set is a collection of items with 2 properties:
a. It doesn’t contain duplicates.
b. The order of the elements does not matter.� (but we usually write the items in ascending order)
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What is a set?
Is it a set?
Are these sets the same?
False
True
False
True
True
True
True
False
False
False
True
False
What is a set?
It’s possible to have a set of sets!
A set with one thing in it is known as a singleton.
Sets need not consist of numbers.
Finite Sets vs Infinite Sets
The examples with seen have been finitely large sets.
But it’s also possible to have sets which are infinitely large…
“the set of all positive integers (whole numbers)”
“the set of all odd numbers”
The Empty Set
We can also have a set with nothing in it!
(think of it as an empty bag)
It is known as the empty set:
We’ll see why it’s useful next lesson.
Fro Fact: This is (as far as I know) the only Scandinavian letter used in maths.
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Venn Diagrams
Venn Diagrams are a way of showing the items in each set.
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(Click to move!)
Example
Test Your Understanding
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? Venn Diagram
Further Examples
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An ‘element’ or ‘member’ is an item in a set.
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True or false?
Elements of Sets and Subsets
False
True
True
False
False
True
True
False
Exercise 1
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#2 :: Venn Diagrams involving Frequencies
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Dr Frost’s cat “Pippin”
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Sometimes we just have one number in each region, representing the number of items in that region, rather than the items themselves.
Fro Tip: The trick is to start from the centre of the diagram and work outwards.
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#2 :: Venn Diagrams involving Frequencies
Conditional Probabilities
Dr Frost’s cat “Pippin”
Given that a randomly chosen person owns a cat, what’s the probability they own a dog?
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Test Your Understanding
The following shows the results of a survey on the types of exercise taken by a group of 100 people.
65 run 48 swim
60 cycle 40 run and swim
30 swim and cycle 35 run and cycle 25 do all three
(a) Draw a Venn Diagram to represent these data. (4)
Find the probability that a randomly selected person from the survey
(b) takes none of these types of exercise, (2)
(c) swims but does not run, (2)
Jason is one of the above group. Given that Jason runs,
(e) find the probability that he swims but does not cycle. (3)
Edexcel S1 Jan 2012 Q6
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Question 1
[JMC 2002 Q11] The Pythagoras School of Music has 100 students. Of these, 60 are in the band and 20 are in the orchestra. Given that 12 students are in both the band and the orchestra, how many are in neither the band nor the orchestra?
Exercise 2
(on provided sheet)
B
Orc
12
48
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32
32 students
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Exercise 2
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(on provided sheet)
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Exercise 2
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(on provided sheet)
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Exercise 2
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(on provided sheet)
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Exercise 2
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(on provided sheet)
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Exercise 2
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(on provided sheet)
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Exercise 2
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(on provided sheet)
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[SMC 2011 Q17] Jamie conducted a survey on the food preferences of pupils at a school and discovered that 70% of the pupils like pears, 75% like oranges, 80% like bananas and 85% like apples.
What is the smallest possible percentage of pupils who like all four of these fruits?
Exercise 2
(on provided sheet)
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#3 Combining Sets
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* Things in A or B also includes things in both. Things in one but not the other is known as “exclusive or”. You do not need to know this!
#3 Combining Sets
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Quickfire Examples
read this as “A and not B”
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Test Your Understanding
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10,12,14,15,16,18
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(on provided sheet)
Matching Game!
Match the set expressions with the indicated regions.
Click an expression below to reveal.
(on provided sheet)
Cardinality of Sets
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Exercise 3
(on provided sheet)
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Exercise 3
(on provided sheet)
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Exercise 3
(on provided sheet)
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