Download Revision Topic 3: Logic, sets and probability

Mathematical Studies Standard Level for the IB Diploma
Revision Topic 3: Logic, sets and probability
Chapter 8: Set theory and Venn diagrams
Basic concepts of set theory
Using set notation is like writing in code or another language. As long as you understand the symbols
and the structure, you can work out the meaning and thus find the solution.
Notation used:
{1, 2, 3, 4, 5}
U
n( A)
the set of numbers 1, 2, 3, 4 and 5
the universal set, which consists of everything that could be included
the number of elements in the set A
∩
intersection; A ∩ B consists of the elements common to sets A and B
⊂
subset; A ⊂ B means that the set A lies within the set B
∈
element; x ∈ A means that x is a member of the set A
∉
not an element; y ∉ A means that y is not a member of the set A
∅
the empty set, a set with no elements
∪
union; A ∪ B means the collective elements of sets A and B when they are
merged together
A′
the complement of the set A, meaning the elements that are not in A
Venn diagrams with numbers
A Venn diagram uses circles to contain the elements of sets.
Elements that are in more
than one set are placed in
the overlap of the circles,
called the intersection.
You may be asked to consider situations involving two or three sets. The following example
summarises the various questions that might arise.
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Let U = {the integers from 0 to 16}
A = {2, 4,6,8,10,12,14,16}
B = {1, 4,9,16}
Some questions can be worked out from the lists of elements of the sets, e.g.
n( A) = 8
6∈ A
5∉ B
A′ ={1,3,5,7,9,11,13,15}
Others benefit from using a Venn diagram:
Question and answer in set notation
A∩ B =
{4,16}
Meaning in words
The overlap between
A and B
1, 2,3,5, 6, 7,8,9, 
( A ∩ B)′ =


10,11,12,13,14,15
Everything not in the
overlap of A and B
A∪ B =
{1, 2, 4, 6, 8, 9,10,12,14,16}
A and B merged
together
( A ∪ B )′ =
{3, 5, 7, 11, 13, 15}
Everything not in the
merged set A and B
A′ ∩ B =
{1, 9}
Everything that is not
in A but is in B
( A′ ∩ B )′
A ∪ B′ =
Everything that is in
A merged with
everything not in B
2, 3, 4, 5, 6, 7,8,10, 
=

11,12,13,14,15,16 
A∪ B − A∩ B =
{1, 2, 6, 8, 9,10,12,14}
Answer as a Venn diagram
Everything in A or B
but not in both
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Questions may be given in context, in which case you should adapt the answer you give so that it
makes sense within the given context.
You may have as many as three sets to
consider. Then you will have three circles
in your Venn diagram.
If one set is completely contained in
another, it is called a subset. In a Venn
diagram, a subset is represented by a
circle inside a larger circle.
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Chapter 9: Logic
Propositions
•
Logic uses statements that can be either true or false.
•
A letter (normally p, q, r or s) is allocated to the statement.
•
When defining the proposition, the letter is followed by a colon and then the statement.
Propositions and compound statements are used to analyse situations and provide reasoning to
validate arguments.
The basic vocabulary and symbols of logic are summarised in the following table, which also makes
links to the set notation equivalents:
Term
Meaning
Symbols used
Proposition
A statement that can be either
true or false
The opposite of a proposition,
i.e. not the proposition
p, q, r, s etc.
Negation
¬p , meaning ‘not p’
In set notation this
is equivalent to
A, a set
A′ , the complement
of the set A
For example, if a proposition is
p: 8 is an even number
then its negation is
¬p : 8 is not an even number.
Compound statements
A compound statement is formed when one proposition is connected with another proposition. It is
like a sentence written in the language of logic. You need to be able to work out what a compound
statement is saying and know how to compare two statements with each other. The following symbols
are used to make compound statements:
In symbols This is called
Meaning
p∧q
p∨q
Both p and q at the same time
Either p or q, or both
In set notation this is
equivalent to
A ∩ B , the intersection
A ∪ B , the union
Either p or q, but not both
A∪ B − A∩ B
p∨q
conjunction
disjunction
exclusive
disjunction
p⇒q
implication
p⇔q
equivalence
One thing leads to another. In English it
is usually stated as ‘p implies q’ or ‘if p
then q’.
Each statement implies the other
statement (like a two-way implies
relationship)
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A ⊂ B , subset
A=B
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You should be able to understand and write sentences using this symbolic notation.
For example, given the propositions
p: x is an even number
q: x can be divided by 4
the statement p ∧ ¬q means that x is an even number that cannot be divided by 4.
Using truth tables
After creating a compound statement, you need to determine whether it is true or false.
In the above example, the compound statement q ⇒ p says that ‘if x can be divided by 4 then x is an
even number’, which is true.
However, if x is an even number, it does not necessarily follow that x can be divided by 4, so the
statement p ⇒ q would be false.
It can be hard to analyse a complex situation containing two or three statements; truth tables can help
wth this.
•
A truth table summarises all the possibilities that could occur and whether each one is true or
false.
•
Each column of the table is headed by a statement or compound statement.
•
Underneath each statement is a list of all the possibilities.
The following truth table is for any two propositions p and q and their standard compound statements.
This table is given in the formula booklet.
p
q
¬p
p∧q
p∨q
p∨q
p⇒q
p⇔q
T
T
F
T
T
F
T
T
T
F
F
F
T
T
F
F
F
T
T
F
T
T
T
F
F
F
T
F
F
F
T
T
not p (the
opposite
of p)
both p
and q
p or q or
both p
and q
p or q but
not both
For this
statement to be
true, if p is
true then q
must also be
true; if p is
false then q
can be either.
p and q
should be
the same
for this to
be true.
These are the four
possible
combinations of
outcomes for the
two propositions.
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Testing logically
There are a range of different positions, situations and arguments that can be tested logically using
truth tables.
You need to know the following vocabulary to test logically:
Logical
equivalence
Tautology
Contradiction
Converse
Inverse
Contrapositive
When two compound statements mean the same thing
When a compound statement is always true
When a compound statement is never true
The reverse of an implication, i.e. the converse of p ⇒ q is q ⇒ p
The negative opposite of an implication, i.e. the inverse of p ⇒ q is ¬p ⇒ ¬q
The inverse converse of an implication, i.e. the contrapositive of p ⇒ q is ¬q ⇒ ¬p
You may be asked to test if a compound statement is a tautology or whether two compound
statements are logically equivalent. Whatever you are asked to test, constructing a truth table will help
you.
For example, test whether the compound statement ( p ∨ q) ∧ ¬q ⇒ p ∨ q is a contradiction, a
tautology or neither.
Construct the truth table containing all the elements you need:
Start with
Work out the two
the two
parts of the LHS
propositions
p∨q
¬q
p
q
T
T
F
F
T
F
T
T
F
T
T
F
F
F
F
T
Then look at the rule for Work out
conjunction of these two the RHS
parts to give the LHS
p∨q
( p ∨ q ) ∧ ¬q
F
T
T
T
F
T
F
F
Then compare to see if
the LHS implies the
RHS
( p ∨ q ) ∧ ¬q ⇒ p ∨ q
T
T
T
T
Because the last column is always true, the compound statement ( p ∨ q) ∧ ¬q ⇒ p ∨ q is a tautology.
For an exam question that asks you to test logically:
•
Break up the compound statement into parts, and decide if each part is true or false.
•
When trying to test if two compound statements are logically equivalent, complete a separate
truth table for each statement and then check whether the final columns of the two tables are
exactly the same.
•
When trying to find out if a compound statement is a tautology, check whether the final
column of the truth table contains only T.
•
When trying to find out if a compound statement is a contradiction, check whether the final
column of the truth table contains only F.
•
Use the truth table in the formula booklet to help you with the true or false combinations.
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Chapter 10: Probability
Introduction to probability
The probability of an event is how likely it is to happen.
The notation for probabilities is P(event). For example, P(live to 100) stands for the probability that
someone will live to 100 years old. You need to be able to read statements written in this notation.
Calculating probability
In situations where we are interested in a single event A, probability is calculated using this formula:
P( A) =
number of outcomes in A
total number of outcomes
Expected value
The expected value is how many times ‘on average’ you think something will happen.
You can find the expected value by the following formula:
Expected value = probability of success × number of trials
Complementary events
You have complementary events when either one thing or another must be true.
•
The probabilities of complementary events must add up to 1.
•
For a pair of complementary events, if one event is A then the other is A′ or ‘not A’.
•
So P( A) = 1 − P( A′) .
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Sample space diagrams
A sample space diagram is a list of all the possible outcomes put into a table or graph format.
Such a diagram helps you work out how many outcomes contribute to a particular event of interest.
The most common diagrams that you might see are summarised in the following table.
Type
Combination
table
Grid
Description
Example
This gives a visual representation Throwing a die and a coin:
of every possible combination of
outcomes. Each combination can
Die
1
2
3
4
5
6
be represented by a pair (as shown Coin
in the example), a total (if the two
Heads
H1 H2 H3 H4 H5 H6
results are both numbers), T or F,
Tails
T1 T2 T3 T4 T5 T6
or some other format appropriate
to the context.
This is most commonly used to
Throwing two dice:
show the outcomes of a combined
event made up of two separate
events. The possible outcomes of
the two individual events are
listed on the horizontal and
vertical axes, and each possible
combination outcome is marked
as a cross or dot on the grid.
Tree
diagram
This is useful for depicting
combined events where one event
occurs after another. The possible
outcomes of each event are shown
as branches. To reach the
combined outcome of interest,
you follow the path formed by the
relevant outcomes.
Finding the probability of getting two sixes when
two dice are thrown:
Simple
frequency
table of
outcomes
This allows for all outcomes to be
assigned a unique probability and
is often used when finding
probabilities using an experiment.
Finding the probability that students would be
late for school on any particular day:
Day of the
week
Number of
late students
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Mon
Tue
Wed
Thu
Fri
18
11
14
7
35
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Type
Two-way
frequency
table
Description
This is often used when you have
overlapping events that involve
the same population.
Example
Finding the probability of adults in a certain age
group staying at a hotel being male or female:
Gender
Male
Female
Total
6
27
16
49
21
81
50
152
Age group
18–25
25–50
50+
Total
Venn
diagram
This is generally useful when you
are considering overlapping (i.e.
not mutually exclusive) events.
15
54
34
103
Finding the probability that a number below 20 is
an even square number:
(Please note known error, ‘6’
should be inside the ‘even’ set i.e.
the blue circle. A replacement
figure will be supplied as soon as
possible, apologies for any
inconvenience caused.)
Probability of combined events
The following rules for calculating the probability of combined events are provided in the formula
booklet:
Combined event
OR
A∪ B
AND
A∩ B
Meaning
A or B
happens
A and B both
happen
Formula
P( A ∪ B )= P( A) + P( B ) − P( A ∩ B )
P( A ∩ B ) =
P( A) P( B ) if A and B are independent events
P( A ∩ B ) =
0 if A and B are mutually exclusive events
Using tree diagrams
When two or more events happen (at the same time or one after another), we can use a tree diagram to
visualise the outcomes of the combined event.
•
The tree branches show all possible outcomes.
•
Following one sequence of branches gives you one of the outcomes of the combined event; to
find the probability of this outcome you multiply together (AND) the probabilities along the
branches.
•
To find the probability of an event that is made up of several different outcomes, you add
together (OR) the probabilities of the different outcomes, assuming there is no overlap
between them.
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For example, suppose there are some numbered balls in a bag. You pick one ball at random, put it
back and then pick another.
Given that P(even) = 0.4, what is the probability that you get one even and one odd ball?
P(even ∩ odd) = 0.4 × 0.6 = 0.24
P(odd ∩ even) = 0.6 × 0.4 = 0.24
P ( (even ∩ odd) ∪ (odd ∩ even) ) = P(even ∩ odd) + P(odd ∩ even) = 0.24 + 0.24 = 0.48
Selection without replacement
If you select an object from a set and don’t put it back (selection without replacement), this changes
the probabilities of future selections from that set.
Calculate the new probabilities based on what has been removed; then use the same method (such as
tree diagrams) to work out probabilities of combined events made up of successive selections.
Conditional probability
A conditional probability is the likelihood of one thing happening given that something else has also
happened.
In the formula booklet you are given this formula for conditional probability:
P(A | B ) =
P( A ∩ B )
P( B )
For the above example of odd- and even-numbered balls, you could use this formula to find that
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Notice that we get 0.4, which is just P(even). Often, you can get the answer quickly by just looking at
the portion of the tree diagram corresponding to the ‘given’ part and then reading the probability from
the appropriate branch, which in this case is the branch joining (first) ‘odd’ and (second) ‘even’:
To calculate conditional probabilities it is common to use a two-way table.
For example, using this table to find
P(aged 50+ | male) , look at just the ‘Male’
column (103 in total) and ignore the rest.
There are 34 in the 50+ age group, so
34
P(aged 50+ | male) =
103
Gender
Age group
18–25
25–50
50+
Total
Male
15
54
34
103
Female
6
27
16
49
Total
21
81
50
152
Using the formula gives the same answer but is more complicated:
P(aged 50+ ∩ male)
P(male)
34
103
=
∩ male)
=
P(aged 50+
and
P(male)
152
152
34 103 34
÷
=
so P(aged 50+ |male) =
152 152 103
P(aged 50+ |male) =
Testing for types of events
You can work out what sort of relationship there is between two events you’re considering by using
the probability formulas as a test.
Type of events
Mutually exclusive events
Combined events
Independent events
This means that
Probability relation
Both events cannot happen
P(A ∪ B)= P( A) + P( B)
at the same time.
Both events could happen at P(A ∪ B) ≠ P( A) + P( B)
the same time.
The two events are unrelated P( A ∩ B)= P( A) × P( B)
and don’t affect one another.
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