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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

• 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

Subset

Intersect

This is a disjoint set

• 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 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 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

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.

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

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:

• 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 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

• 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: 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)