Transcript Review Topic 3 PowerPoint II

IB Math Studies – Topic
3
IB Course Guide Description
IB Course Guide Description
Notation
Symbol
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: 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
p  q is same as P  Q
Q
P
in easier terms p  q means that
q is true whenever p is true
Equivalence
• Two statements are equivalent if one of the
statements imples the other, and vice versa.
– p if and only if q
• pq
Q
p  q is same as P = Q
P
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
success
P(E) 
total
•
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.
P(A and B) = P(A) × P(B)
–
And = Multiplication
• Dependent: Events where the occurrence of
one of the events does affect the occurrence of
the other event.
P(A then B) = P(A) × P(B given that A has occurred)
Laws of probability
Sampling with and without
replacement