IB Math Studies – Topic 3

IB Course Guide Description

Notation SymbolNotation ⊆ Subset ∈ Is an element of ∉ Is not an element of ∪ Union ∩Intersect

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.

Venn Diagrams - representation:

Compound Propositions Compound Propositions are statements that use connectives and and or, 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.

Venn Diagram – representation

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 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 p  q is same as P  Q

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 p  q is same as P = Q

Summary of Logic Symbols

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

Venn Diagrams

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(B given that A has occurred)

Laws of probability

Sampling with and without replacement