ib math studies – topic 3
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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 PresentationTRANSCRIPT
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IB Math Studies – Topic 3
Sets, Logic and Probability
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IB Course Guide Description
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IB Course Guide Description
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NotationSymbol Notation
⊆ Subset
∈ Is an element of
∉ Is not an element of
∪ Union
∩ Intersect
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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
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Venn Diagrams Subset
Intersect
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Union
This is a disjoint set
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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.
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• Venn Diagrams - representation:
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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.
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• Venn Diagram – representation
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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
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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.
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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
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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
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Summary of Logic Symbols
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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
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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)
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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
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Venn Diagrams
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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)
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Laws of probability
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Sampling with and without replacement