Topic 3 Sets, Logic and Probability

Joanna Livinalli and Evelyn Anderson

IB Course Guide description

Topic 3.1 and 3.3: Set Theory

Set Notation

Symbol Definition Means…

U universal set 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 An overlap; only the elements in both sets.

Union A marriage; all the elements in each set, including all elements that are in both.

Subset A smaller set contained wholly within a larger set

n(A) Number of elements inside set A

Topic 3.2:Venn diagrams and sets

Intersection

Union Subset

Mutually exclusive events

(A B)’

A’ B

(A B)’

A’ B

Example Problem 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.

Check your answers (a)

(b) There 4 students who do Spanish only (c) Shade everything but the union of E and F

(where the 6 is)

Topic 3.4 – 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 ¬.

• Conjunction – the word ‘and’ used to join two conjunctions together. For example, the baby is a girl and Australian. The symbol used is .

p ¬p

T F

F T

p q p qT 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 .

• 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”. The symbol used is

p q p q

T T T

T F T

F T T

F F F

p q p q

T T F

T F T

F T T

F F 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 .

• 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 q pqT T T

T F F

F T T

F F T

p q pq

T T T

T F F

F T F

F F T

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.

• Tautology – a statement that produces True (T) throughout the column of the truth table.

Column (?) would be a tautology

• Contradiction – a statement that produces False (F) throughout the column of a truth table.

Column (?) Would be a contradiction

p q ?T T T

T F T

F T T

F F T

p q ?T T F

T F F

F T F

F F F

Example Problem

P Q r p q (p q) (p q) rT T T T T F T F T T F F F T T F T F F F T F F F

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.

(d) Is (p q) r a tautology, contradiction or neither?

Check your answers (a) (p q) r

(b) If Andrea does not study English then she will not study Spanish

(c)

(d) Neither a tautology nor a contradiction

p q r p q (p q) (p q) rT T T T F TT T F T F TT F T F T TT F F F T FF T T F T TF T F F T FF F T F T TF F F F T F

Probability

Probability ranges between 0 and 1

or

0 and 100%

Terminology

• Trials—times experiment is repeated• Outcomes—different possible results• Frequency—the number of times an outcome

is repeated• Relative frequency—the number of times an

outcome is repeated in terms of a fraction or percentage

Example Problem

• In assigning 32 assignments, Ms. Boswell has noticed that Maxine has done her homework 12 times.

• Describe the trials, outcomes, frequency, and relative frequency of this situation.

Note: The greater the number of trials, the more reliable the information becomes, and the closer the relative frequency will be as a predictor for future outcomes.

3.8 Equally Likely Events

• If the probability of an event A given by P(A)=n(a)/n(U)

• Probability of a complementary event isP(A’)=1-P(A)

• Example: In a coin, there are two possible outcomes—heads or tails. So the probability of it not being tails is equal to 1-(the probability of it being heads, or .5), which is also .5.

Venn DiagramsThere are 48 students. 22 watch Dragon Ball Z, 15 watch Pokémon, 20 watch Naruto, and 12 watch none.

5

3

What is the probability of (DUP)’? DUP=(30/48)=(5/8)(DUP)’=1-(5/8) or (3/8)What is the probability that the student watches Dragon Ball Z? (22/48) or (11/22)What is the probability the student watches Pokémon and Naruto, but not Dragon Ball Z? (3/48)What is the probability the student watches Naruto or Pokemon, but not both? ((7+6)+(3+5))/48 or 21/48=(7/16)

What is the probability the coin will fall on heads three times in a row? (Tree Diagrams)

Heads (1/2)

Tails (1/2)

Tails (1/2)

Heads (1/2)

Tails (1/2)

Heads (1/2)

(1/2)*(1/2)*(1/2)=(1/8)

Number of possibilities for a particular outcome/Number of total possible outcomes

If there are 5 red marbles and 2 blue marbles in a bag, what is the probability that when

picking one out, one will be blue and the other red? (the marbles are not replaced when

removed)

Red (5/7)

Blue (2/7)

Blue (2/6)

Red (4/6)

Blue (1/6)

Red (5/6)

((5/7)*(2/6))+((2/7)+(5/6))=(10/21)

Table of Outcomes

• Example: Rolling two pairs of dice

http://www.cimt.plymouth.ac.uk/projects/mepres/book8/bk8i10/bk8_10i3.htm

Mutually Exclusive Events

• A situation where only one outcome is possible (i.e., if the coin is heads, it cannot be tails also.)

• A die is tossed. What is the probability the shown is 4?

• What is the probability that it is 4 or 3?• What is the probability that it is 4 and 3?• What is the probability that the dice will roll a 4,

and when tossing a coin at the same time, it will be heads?