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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: 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
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)
