Topic 3 Revision Notes
Sets, Logic & Probability
IB Math Studies
Topics 3.1 & 3.3 Set Theory
Definition
universal set
Set Notation
Means…
All the elements given in a question

is an element of
Is an element or member of the set

is not an element of
Is not an element or member of the set
‘
Complement
opposite of the set

the empty set
an empty set with no members/elements

Intersection

Union

Subset
An overlap; only the elements in both sets.
A marriage; all the elements in each set, including all
elements that are in both.
A smaller set contained wholly within a larger set
Symbol
U
n(A)
Number of elements inside set A
Topic 3.2 Venn diagrams and sets
Often probability questions can be answered by drawing a Venn diagram to represent the problem. Below are
some sample Venn diagrams
Union
Intersection
Subsets
Mutually Exclusive Events
(A  B)’
A’  B
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Topic 3 Revision Notes
Sets, Logic & Probability
(A  B)’
IB Math Studies
A’  B
Guided example
A group of 40 IB students were surveyed about the languages they have chosen at IB: E = English, F =
French, S = Spanish.
3 students did not study any of the languages above.
2 students study all three languages
8 study English and French
10 study English and Spanish
6 study French and Spanish
13 students study French
28 students study English
(a) Draw a Venn diagram to illustrate the data above. On your diagram write the number in each set.
(b) How many students study only Spanish?
(c) On your diagram shade (E  F)’ , the students who do not study English or French.
Answer (a)
The key point here is that when it says, “8 study English and French” it did NOT say, “8 study English and
French only.” So 2 the students that study all three languages have been counted again in the 8 who study
English and French. Hence, only 6 students study English and French only.
• The final diagram will look like this:
Answer (b)
By looking at the Venn diagram we can clearly see that we have 4 who do Spanish only.
Answer (c)
By shading E  F we would have:
Therefore (E  F)’ will be the opposite of this shading.
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Topic 3 Revision Notes
Sets, Logic & Probability
IB Math Studies
Topics 3.4 through 3.7 Logic and truth tables
Propositions – A statement that can be either true or
false. For example, the baby is a girl.
Negation – A proposition that has become negative:
For example, the baby is not Australian. Be careful
that it is a negative and not the opposite. The
symbol used is ¬.
p
¬p
T
F
F
T
Conjunction – the word ‘and’ used to join two
conjunctions together. For example, the baby is a
girl and Australian. The symbol used is .
p
q
pq
T
T
T
T
F
F
F
T
F
F
F
F
Disjunction – the word ‘or’ used to join the
conjunctions together. For example, the baby is a
girl or Australian. The symbol used is .
p
T
T
F
F
q
T
F
T
F
Exclusive Disjunction - is true when only one of the
propositions is true. The word ‘or’ is used again,
but in this case you must write “or, but not both”
For example,
p: Sally ate cereal for breakfast
q: Sally ate toast for breakfast
p
T
T
F
F
q
T
F
T
F
pq
F
T
T
F
Implication – Using the words if …… then with two
propositions. For example: If you do not sleep
tonight then you will be tired tomorrow. The
symbol used for implication is .
p
q
p  q
T
T
T
T
F
F
F
T
T
F
F
T
pq
T
T
T
F
By using the implication we can generate the converse, inverse, and contrapositive.
For example,
p: My shoes are too small.
q: My feet hurt.




Implication: p  q. If my shoes are too small, then my feet hurt.
Converse: q  p. If my feet hurt, then my shoes are too small.
Inverse: p  q. If my shoes are not too small, then my feet do not hurt.
Contrapositive: q  p. If my feet do not hurt, then my shoes are not too small.
Equivalence – If two propositions are linked with “… if and only if…”, then it is an equivalence.
p  q is the same as p  q and q  p. Or using notation: (p  q) = (p  q)  (q  p)
p
T
T
F
F
q
T
F
T
F
pq
T
F
F
T
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Topic 3 Revision Notes
Sets, Logic & Probability
IB Math Studies
Tautology – a statement that produces True (T) throughout the column of the truth table.
Contradiction – a statement that produces False (F) throughout the column of a truth table.
Guided example
Let the propositions p, q, and r be defined as:
p: Andrea studies IB English.
q: Andrea studies IB Spanish.
r: The school offers at least 2 IB languages.
(a) Write the following in logical form. If Andrea studies English and Spanish, then the school
offers at least 2 languages.
(b) Write the following statement in words: ¬p  ¬q
(c) Copy and complete the truth table below.
P
Q
r
p  q (p  q) (p  q) r
T
T
T
T
T
F
T
F
T
T
F
F
F
T
T
F
T
F
F
F
T
F
F
F
(d) Is (p  q) r a tautology, contradiction or neither?
Answer (a)
 The words if … then are in the statement so it must be: (p  q) r
Answer (b)
 If Andrea does not study English then she will not study Spanish.
Answer (c)
 Although these truth tables can appear daunting at first, taking each column one at a time makes
things quite simple.
 The final truth table will be:
p
q
r
p  q (p  q) (p  q) r
T
T
T
T
F
T
T
T
F
T
F
T
T
F
T
F
T
T
T
F
F
F
T
F
F
T
T
F
T
T
F
T
F
F
T
F
F
F
T
F
T
T
F
F
F
F
T
F
Answer (d)
 (p  q) r is neither a tautology nor a contradiction because the final column is neither all T’s
or all F’s.
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Topic 3 Revision Notes
Sets, Logic & Probability
IB Math Studies
Topics 3.8 to 3.10 Probability
Suppose a fair die is rolled. Then there are six possible outcomes, 1
2
3
4
5
6
Suppose you want to know the likelihood of getting a 3. There is only one way of getting a 3 so P (3) 
1
.
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The list of all possible outcomes is called the sample space, and those outcomes which meet the particular
requirement are called the event.
The following formulae are given in the IB formula booklet
n( A)
n(U )
Probability of an event A:
P( A) 
Complementary events:
P(A’) = 1 – P(A)
Combined events:
P( A  B)  P( A)  P( B)  P( A  B)
Mutually exclusive events:
P( A  B)  P( A)  P( B)
Independent events
P( A  B)  P( A) P( B)
Conditional probability
P  A B 
P( A  B)
P( B)
where n(A) is the number of elements in the set A, and n(U) is the number of elements in the whole sample
space U.
Probability of an event
 The probability of event must be written as a number between 0 and 1, and written as a fraction,
percentage or decimal. It is easiest to use a fraction, if possible.
 The first formula means put the number of times A occurs as the numerator and the number of all the
events as the denominator.
Complementary events

If the probability of A is
1
1 2
then the probability of A’ (not A) is 1 – = .
3
3 3
Combined events
 Combined events are events that CAN happen at the same time (for instance choosing a face card or a
heart from a standard deck of playing cards).
 This formula can be understood by using a Venn diagram. Add up the number of times A occurs with
the number of times B occurs and subtract the number of times both occur together.
 For EXAMPLE: in a class of 140 students if 80 people chose Biology, 60 chose Chemistry and 20 chose
both Biology and Chemistry, then P(AB) =

80
60
20 120
.



140 140 140 140
This can be shown with a Venn diagram:
Bio
Ch
60
20
40
20
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Topic 3 Revision Notes
Sets, Logic & Probability
IB Math Studies
GUIDED EXAMPLE
Given P(A) = 0.55, P( A  B)  0.7 and P ( A  B )  0.2 , find P(B’).
SOLUTION:
 P(B’)=1 – P(B)
P( A  B)  P( A)  P( B)  P( A  B)
P(B’) = 1 – 0.35 = 0.65
0.7 = 0.55 + P(B) – 0.2
0.35 = P(B)
Mutually Exclusive events
 Mutually exclusive events are events that CANNOT happen at the same time (for instance choosing a 7
or a face card from a standard deck of playing cards).
 You can think of mutually exclusive events as being a special case of combined events where
P(A∩B)=0.
 So the combined events formula P(AB)=P(A)+P(B) – P(AB) becomes P(AB)=P(A)+P(B).
Much of probability is based on set notation. Looking at the Venn diagram below we can see the numbers that
represent the various parts of the formulae
A
B
10
20
15
5
using this information we can give the following probabilities:
P(A) =
30
50
P(A’) = 1 
30 20

50 50
P(AB)=
30 35 20 45



50 50 50 50
P(A│B) =
20
35
P(B│A) =
20
30
Independent events



The term “independent” is equivalent to “statistically independent.” Statistically independent is similar,
although not exactly, as the everyday usage of the word ‘independent’ in the sense that independent
events are events that have nothing to do with each other. For EXAMPLE, the event of flipping a coin
and tossing a die.
P(A∩B)=P(A)P(B) for independent events.
By definition the following is true for independent events: P(A|B)=P(A)=P(A|B′).
Conditional probability


This is used when we see the ‘given that’ in the question. P(A/B) means the probability of A given B.
When we are ‘given’ something in probability questions we can cut down the outcomes we have to only
things where B has occurred. This forms the denominator of the fraction. The numerator of the fraction
is where A and B have occurred.
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Topic 3 Revision Notes

Sets, Logic & Probability
IB Math Studies
An easy way to answer the “given that” questions is with a Venn diagram, if possible. Whatever
follows the “given that” becomes the sample space or the bottom of the fraction.
GUIDED EXAMPLE
An unbiased 6-sided die is marked with the numbers: 2, 2, 3, 4, 5, and 8.
The die is rolled. Find the probability of the die:
(a) landing on a prime
(b) not landing on a 4
(c) landing on a 2 given that it has landed on a prime.
ANSWER
Probability from tables
Frequently probability questions are often given in tables. These questions are reasonably easy to answer. The
conditional probability question is the only one that sometimes confuses students.
EXAMPLE
The table below shoes the choices made by males and females for their IB second language.
Female
Male
Total
French
60
20
80
Spanish
50
70
120
Total
110
90
200
A student is chosen at random. Find the probability that the student is:
(a) female,
(b) male and studies French,
(c) a female, given that the student studies Spanish.
ANSWER
(a) The total is 200 so that will become the denominator of the fraction. There are 110 females so the
answer is:
110
.
200
(b) There are 20 students who are male and study French so the answer is
20
200
(c) The ‘given that’ makes this question a conditional probability question. For the denominator we
just need the total number of students who study Spanish. The numerator would be the number of
females who study Spanish. So the answer is:
50
= 0.417.
120
The problem above can also be answered by using the formula P ( A B ) 
P( A  B)

P( B)
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Topic 3 Revision Notes
Sets, Logic & Probability
IB Math Studies
50
P( female and studies Spanish )
 P( female Spanish )  200
P( female Spanish ) 
P( Spanish )
120
200
Tree diagrams
Tree diagrams are a clear way to view all the outcomes and their associated probabilities when 2 or more events
occur. From a tree diagram you can find simple probabilities or even a conditional probability.
EXAMPLE
1.
When Geraldine travels to work she can travel either by car (C), bus (B) or train (T). She travels by car on
L) when travelling by
car, bus or train are 0.05, 0.12 and 0.08 respectively.
(a)
Complete the tree diagram below, where NL represents not late.
L
C
0.2
NL
L
B
NL
L
T
NL
a) Find the probability that Geraldine travels by bus and is late.
b) Find the probability that Geraldine is late.
c) Find the probability that Geraldine travelled by train, given that she is late.
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