IB Math Studies – Topic 3

Sets, Logic and Probability

IB Course Guide Description

IB Course Guide Description

Set Theories

•A set is any collection of things with a common property: it can be finite. ▫Example: set of students in a class

•If A= {1,2,3,4,5} then A is a set that contains those numbers

Subsets▫If P & Q are sets then:

P Q means ‘P is a subset of Q’ In every element in P is also an element in Q

Complements▫If A contains elements of even numbers

then A’ contains elements of odd numbers.

Union and Intersection•PQ is the union of sets P and Q meaning all elements

which are in P or Q. •PQ is the intersection of P and Q meaning all

elements that are in both P and Q.▫Example:

•A = {1,2,3,4,5} B= {2,4,6,7}AB = {1,2,3,4,5,6,7}AB = {2,4}

Venn Diagrams• Venn Diagrams are diagrams used to represent sets of objects,

numbers or things. • The universal set is usually represented by a rectangle whereas

sets within it are usually represented by circles or ellipses.

Sets within Venn Diagram

SubsetBA

IntersectionAB

UnionAB

Disjoint or Mutually Exclusive sets

Logic• Proposition

▫The building block of logic. ▫This is a statement that can have one of the two

value, true or false.• Negation• The negation of a proposition is formed by putting

words such as “not” or “do not.”• The negation of a proposition p is “not p” and is written as

¬p. For example:

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

Truth Tables

•A truth table shows how the values of a set of propositions affect the values of other propositions.

•A truth table for negation

p ¬ p

T F

F T

Compound Propositions•Conjunction: The word and can be used to

join two propositions together. Its symbol is .

•Disjunction: The word or can also be used to join propositions. Its symbol is .

•Exclusive Disjunction: Is true when only one of the propositions is true.

•This symbol means that its either p or q but not both. p q

Conjunction/Disjunction and Venn Diagrams

Tautology

•A tautology is a compound proposition that is always true, whatever the values of the original propositions.

•Example:

•When all the final entries are ‘T’ the proposition is a Tautology.

p ¬p p¬p

T F T

F T T

Contradiction

•A Contradiction is a compound proposition that is always false regardless of the truth values of the individual propositions.

•Example:

•When all the final entries are ‘F’ the proposition is a Contradiction.

p ¬p p¬p

T F F

F T F

Logically Equivalent• If two statements have the same truth tables then they

are true and false under the exact same conditions. •The symbol for this is •The wording would be said as: “if and

only if”p q pq ¬(pq)

T T T F

F T F T

T F F T

F F T F

¬p ¬q ¬p¬q

F F F

T F T

F T T

T T F

Implication

• If two propositions can be linked with “If…, then…”, then we have an implication.

•p is the antecedent and q is the consequent•The symbol would be •For example:

▫P: You steal▫Q: you go to prisonTherefore, the words “If” and “then” are added. “if you steal, then you go to prison.”

Converse, Inverse, and Contrapositive

•Converse: q p•Inverse: p q•Contrapositive: q p

▫For example: ▫P: It is raining▫Q: I will get wet

Converse: “If it is raining, then I will get wet.” Inverse: “It it isn’t raining, then I won’t get wet.” Contrapositive: “It I’m not wet, then it isn’t raining.”

Probability•Probability is the study of the chance of events

happening

total

successP(E)

Combined Events•The probability of event A or event B

happening. P(AB) = P(A)+P(B)

•However, this formula only works with A and B are mutually exclusive (they cannot happen at the same time)

•If they are not mutually exclusive, use: P(AB)= P(A)+P(B)-P(AB)

Sample Space

• There are various ways to illustrate sample spaces:• Sample space of possible outcomes of tossing a coin.

• Listing• Sample space = {H,T}

• 2-D Grids

• Tree Diagram

Theoretical Probability•A measure of the chance of that event

occurring in any trial of the experiment. •The formula is:

Using Tree Diagrams

Tree Diagrams – Sampling• Sampling is the process of selecting an object from a large

group of objects and inspecting it, nothing some features • The object is either put back (sampling with replacement)

or put to one side (sampling without replacement).

Laws of ProbabilityType Definition Formula

Mutually Exclusive Events

Events that cannot happen at the same time

P(A ∩ B) = 0 P(A B) = P(A) + P(B)

Combined Events (a.k.a Addition Law)

Events that can happen at the same time

P(AB) = P(A) + P(B) – P(A∩B)

Conditional Probability

The probability of an even A occurring, given that event B occurred.

P (A | B) = P (A ∩ B) P (B)

Independent Events Occurrence of one event does NOT affect the occurrence of the other

P(A ∩ B) = P(A) P(B)