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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} - PowerPoint PPT Presentation
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IB Math Studies – Topic 3
Sets, Logic and Probability
IB Course Guide Description
IB Course Guide Description
NotationSymbol Notation
⊆ 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: 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• 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 trueP
Qp 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
totalsuccess
P(E)
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
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