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IB Math Studies – Topic 3

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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

Union

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.

- Venn Diagrams - representation:

- 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

- For example: Pierre listens to dubstep and rap

- ‘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 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

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 q

- Hence if p then q

P

- p q is same as P Q

Q

P

- Two statements are equivalent if one of the statements imples the other, and vice versa.
- p if and only if q
- p q

- p if and only if 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

Venn Diagrams

- 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)