IB Math Studies – Topic 3

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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

Sets, Logic and Probability

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.
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.

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

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

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

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
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
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

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)