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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

IB Math Studies – Topic 3

Sets, Logic and Probability

slide5
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
Venn Diagrams

Subset

Intersect

slide7

Union

This is a disjoint set

logic
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
  • 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 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
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
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
equivalence

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, 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
  • 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
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 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)

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