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IB Math Studies – Topic 3. Sets, Logic and Probability. IB Course Guide Description. IB Course Guide Description. Notation. Sets. Infinite Sets: These are sets that have infinite numbers. Like {1,2,3,4,5,6,7,8,…} F inite Sets: These are sets that finish. Like {1,2,3,4,5}

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IB Math Studies – Topic 3

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Ib math studies topic 3

IB Math Studies – Topic 3

Sets, Logic and Probability


Ib course guide description

IB Course Guide Description


Ib course guide description1

IB Course Guide Description


Notation

Notation


Ib math studies topic 3

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

Venn Diagrams

Subset

Intersect


Ib math studies topic 3

Union

This is a disjoint set


Logic

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.


Ib math studies topic 3

  • Venn Diagrams - representation:


Compound propositions

Compound Propositions

  • Compound Propositions are statements that use connectives andandor, 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.


Ib math studies topic 3

  • Venn Diagram – representation


Inclusive and exclusive disjunction

Inclusive and Exclusive Disjunction

  • Inclusive disjunction: is true when one or both propositions are true

    • Denoted like this: pq

    • 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: pq

    • Said like: p or q but not both


Truth tables

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

Implication

Q

  • 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 true

P

  • p  q is same as P  Q


Equivalence

Q

P

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 is same as P = Q


Summary of logic symbols

Summary of Logic Symbols


Converse inverse and contrapositive

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

  • 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


Sample space

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


Ib math studies topic 3

Venn Diagrams


Independent and dependent events

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(Bgiven that A has occurred)


Laws of probability

Laws of probability


Sampling with and without replacement

Sampling with and without replacement


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