IB Math Studies – Topic 3

IB Math Studies – Topic 3

Sets, Logic and Probability

IB Course Guide Description

IB Course Guide Description

NotationSymbol Notation

⊆ Subset

∈ Is an element of

∉ Is not an element of

∪ Union

∩ Intersect

Sets• Infinite Sets: These are sets that have

infinite numbers. Like {1,2,3,4,5,6,7,8,…}

• Finite Sets: These are sets that finish. Like {1,2,3,4,5}

• Some sets however don’t have anything, these are empty sets. n( ) = 0

Venn Diagrams Subset

Intersect

Union

This is a disjoint set

Logic• Propositions: Statements which can either be true or

false– These statements can either be true, false, or indeterminate.– Propositions are mostly represented with letters such as P, Q

or R• Negation: The negation of a proposition is its negative.

In other words the negation of a proposition, of r, for example is “not r” and is shown as ¬r.

Example:p: It is Monday.¬p: It is not Monday.

• Venn Diagrams - representation:

Compound Propositions• Compound Propositions are statements that

use connectives and and or, to form a proposition. – For example: Pierre listens to dubstep and rap• P: Pierre listens to dubstep• R: Pierre listens to rap

– This is then written like: P^R• ‘and’ conjunction – notation: p q

• ‘or’ disjunction– notation: p q

Only true when both original propositions are true

p q is true if one or both propositions are true.

p q is false only if both propositions are false.

• Venn Diagram – representation

Inclusive and Exclusive Disjunction• Inclusive disjunction: is true

when one or both propositions are true• Denoted like this: pq• It is said like: p or q or

both p and q• Exclusive disjunction: is only

true when only one of the propositions is true• Denoted like this: pq• Said like: p or q but not

both

Truth Tables

A tautology is a compound statement which is true for all possibilities in the truth table.

A logical contradiction is a compound statement which is false for all possibilities in the truth table.

Implication• An implication is formed using “if…then…”– Hence if p then q• p q

in easier terms p q means that

q is true whenever p is trueP

Qp q is same as P Q

Equivalence• Two statements are equivalent if one of the

statements imples the other, and vice versa.– p if and only if q• p q

P

Q p q is same as P = Q

Summary of Logic Symbols

Converse, Inverse, and Contrapositive

• Converse:– the converse of the statement p q is q p

• Inverse:– The inverse statement of p q is p q

• Contrapositive:– The contrapositive of the statement p q is q p

Probability• Probability is the study of the chance of events happening.• An event which has 0% change of happening (impossible) is

assigned a probability of 0• An event which has a 100% chance of happening (certain) is

assigned a probability of 1– Hence all other events are assigned a probability between

0 and 1

totalsuccess

P(E)

Sample Space• There are many ways to find the set of all possible outcomes of an experiment. This is the

sample space

Tree Diagram

Dimensional Grids

Venn Diagrams

Independent and dependent events

• Independent: Events where the occurrence of one of the events does not affect the occurrence of the other event.

– And = Multiplication

• Dependent: Events where the occurrence of one of the events does affect the occurrence of the other event.

P(A and B) = P(A) × P(B)

P(A then B) = P(A) × P(B given that A has occurred)

Laws of probability

Sampling with and without replacement