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Set Notation & Venn Diagrams (OCR GCSE Maths)
Revision Note
Set Notation
What is a set?
- A set is a collection of elements
- Elements could be anything - numbers, letters, coordinates
- You could describe a set by writing its elements inside curly brackets { }
- e.g. {1, 2, 3, 6} is the set of the factors of 6
- In probability a subset of possible outcomes can be useful
- e.g. If a whole number between 1 and 50 is chosen at random
- we might be interested in the outcome being a multiple of 10
- the set {10, 20, 30, 40, 50}
- which is a subset of the set {1, 2, 3, ..., 48, 49, 50}
- e.g. If a whole number between 1 and 50 is chosen at random
What do I need to know about set notation?
- You do not need to remember any of the notation in the notes below
- but you may see them used in textbooks, on websites, etc
- so it is helpful to be aware of them, or have somewhere to look them up
- is the universal set (the set of everything)
- e.g. if talking about factors of 24 then = {1, 2, 3, 4, 6, 8, 12, 24}
- You may see alternative notations used for
- U is a common alternative
- S or the Greek letter ξ (xi) may also be seen
- Sets, particularly in probability, are often referred to by capital letters
- e.g. The set A is the set of even factors of 24
- A = {2, 4, 6, 8, 12, 24}
- a ∈ B means a is an element of B (a is in the set B)
- A ∩ B means the intersection of A and B (the overlap of A and B)
- the set of elements that are in both sets A and B
- in probability, intersection (∩) generally means AND
- A ∪ B means the union of A and B (everything in A or B or both)
- the set of elements that are in either set A, or set B, or both
- in probability, union (∪) generally means OR
- A' (called "A prime" but frequently called "A dash") means "not A" (everything outside A)
- the set of elements that are not in set A
- You may occasionally see the symbol ∅ - this is the empty set (the set with no elements)
Venn Diagrams
What is a Venn diagram?
- Venn diagrams allow us to show two (or more) characteristics of a situation where there is overlap between the characteristics
- For example, students in a sixth form can study biology or chemistry but there may be students who study both
What might I be asked to do with a Venn diagram?
- You can be asked to
- draw a Venn diagram and/or
- interpret a Venn diagram
- Strictly speaking the rectangle (box) is always essential on a Venn diagram
- it represents everything that can happen in the situation
- you may see the letter or written inside or just outside the box
- this means “the set of all possible outcomes” - i.e. “everything”!
- sometimes the letters U or S are used instead
- The words AND and OR become very important in both drawing and interpreting Venn diagrams
- You will need to be familiar with the terms
- intersection (means AND)
- union (means OR)
How do I draw a Venn diagram?
- Start with a “box” and overlapping “bubbles”
- the number of bubbles needed will depend on how many characteristics you are dealing with
- it will usually be 2 or 3
- Work through each sentence/piece of information given in a question to begin completing sections of your Venn diagram
- pieces of information may have to be combined before you can enter a value into the diagram
- not all values will be given directly
- some may need working out
- you will be expected to do this to complete your Venn diagram
- Remember to consider AND and OR
How do I interpret a Venn diagram?
- Use the information in the question to identify the parts of the Venn diagram needed to answer it
- Shading the relevant parts of a Venn diagram can be helpful
- Be careful with
- probability notation such as A' (not A)
- terms such as intersection (AND) and union (OR)
- If you look at questions from other boards or older papers then you might see the following notation ∩ for intersection and ∪ for union
- You do not need to know these for your exam - words will be used
How do I use Venn diagrams with conditional probability?
- Probabilities that only involve a subset of the things in a Venn diagram are called conditional probabilities
- For example with students studying German or Spanish or both, you might want to know the probability that a student who studies Spanish also studies German
- This would require considering only the 'Spanish' part of the Venn diagram when finding the probability
- Conditional probability questions are often (but not always!) introduced by the expression 'given that...'
- For example 'Find the probability that a randomly chosen student studies German, given that the student also studies Spanish'
- The answer would be the number in the 'Spanish AND German' part of the diagram divided by the sum of all the numbers in the 'Spanish' part of the diagram
- Conditional probabilities are sometimes written using the 'straight bar' notation
- That is read as 'the probability of A given B'
- For example would be the probability that a student studies German, given that the student also studies Spanish
Examiner Tip
- You may have to use your Venn diagram more than once in a question
- so shading the original diagram can become confusing if you're trying to use it more than once
- draw a 'mini'-Venn diagram (a small quick sketch just showing the box and bubbles but no values) and shade that
Worked example
In a class of 30 students, 15 students study Spanish, and 3 of the Spanish students also study German.
7 students study neither Spanish nor German.
Draw a Venn diagram to show this information.
We start with the 3 in the intersection ("overlap"); we can then deduce the "Spanish only" section is 12.
7 needs to be outside both bubbles but within the box.
With a total of 30 we can work out how many students study "German only" and complete the diagram.
Use your Venn diagram to find the probability that a student, selected at random from the class, studies Spanish but not German.
Highlight the part "Spanish only".
Pick out the numbers you need carefully.
Students studying "Spanish only" = 12
Total number of students = 30
P(Spanish only)
Worked example
Given the Venn diagram below, find the following probabilities:
P(A)
Draw a 'mini'-Venn diagram - a quick sketch without the details.
See these questions as "ways to win" - so in this part you win if in "bubble A".
B and C do not come into it at all.
Total in A = 3 + 5 + 1 + 8 =17
Total = 3 + 5 + 1 + 8 + 2 + 4 + 8 + 9 = 40
P(A and B and C )
(intersection means AND) - "win" if "in A" AND "in B" AND "in C".
Total in A AND B AND C = 8
Total = 40
P(B' and C )
(intersection means AND) - "win" if "not in B" AND "in C".
Total in "not B" AND C = 5
Total = 40
P(A or B )
(UNION means OR) - "win" if "in A" OR "in B".
Total in A OR B = 3 + 1 + 8 + 5 + 2 + 8 = 27
Total = 40
P(A or B or C )
(union means OR) - "win" if "in A" OR "in B" OR "in C".
In this case, it will be easier to subtract from the whole total.
Total in A OR B OR C = 40 - 9 = 31
Total = 40
P(A' or B' )
(union means OR) - "win" if "not in A" OR "not in B".
This one is particularly difficult to see without a diagram!
Total in A' OR B ' = 40 - 8 - 5 = 27
Total = 40
Worked example
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