Transcript IB Math Studies * Topic 3

IB Math Studies – Topic 3
IB Course Guide Description
IB Course Guide Description
Set Theories
• A set is any collection of things with a common
property: it can be finite.
▫ Example: set of students in a class
• If A= {1,2,3,4,5} then A is a set that contains
those numbers
Subsets
▫ If P & Q are sets then:
 P ⊆Q means ‘P is a subset of Q’
 In every element in P is also an element in Q
Complements
▫ If A contains elements of even numbers then A’
contains elements of odd numbers.
Union and Intersection
• P∪Q is the union of sets P and Q meaning all
elements which are in P or Q.
• P∩Q is the intersection of P and Q meaning all
elements that are in both P and Q.
▫ Example:
• A = {1,2,3,4,5}
B= {2,4,6,7}
A∪B = {1,2,3,4,5,6,7}
A∩B = {2,4}
Venn Diagrams
• Venn Diagrams are diagrams used to represent sets of objects, numbers or
things.
• The universal set is usually represented by a rectangle whereas sets within
it are usually represented by circles or ellipses.
Sets within Venn Diagram
Subset
B⊆A
Union
A∪B
Intersection
A∩B
Disjoint or
Mutually
Exclusive
sets
Logic
• Proposition
▫ The building block of logic.
▫ This is a statement that can have one of the two value,
true or false.
• Negation
• The negation of a proposition is formed by putting
words such as “not” or “do not.”
• The negation of a proposition p is “not p” and is written
as ¬p.
 For example:
 p: It is Monday.
 ¬p: It is not Monday.
Truth Tables
• A truth table shows how the values of a set of
propositions affect the values of other
propositions.
• A truth table for negation
p
¬p
T
F
F
T
Compound Propositions
• Conjunction: The word and can be used to
join two propositions together. Its symbol is ∧.
• Disjunction: The word or can also be used to
join propositions. Its symbol is ∨.
• Exclusive Disjunction: Is true when only one of
the propositions is true.
• This symbol p  q means that its either p or q but not
both.
Conjunction/Disjunction and Venn Diagrams
Tautology
• A tautology is a compound proposition that is
always true, whatever the values of the original
propositions.
• Example:
p
¬p
p∨¬p
T
F
T
F
T
T
• When all the final entries are ‘T’ the proposition is a
Tautology.
Contradiction
• A Contradiction is a compound proposition that is
always false regardless of the truth values of the
individual propositions.
• Example:
p
¬p
p∧¬p
T
F
F
F
T
F
• When all the final entries are ‘F’ the proposition is a
Contradiction.
Logically Equivalent
• If two statements have the same truth tables then they
are true and false under the exact same conditions.
• The symbol for this is ↔
• The wording would be said as: “if and only if”
p
q
p∧q
¬(p∧q)
¬p
¬q
¬p∨¬q
T
T
T
F
F
F
F
F
T
F
T
T
F
T
T
F
F
T
F
T
T
F
F
T
F
T
T
F
Implication
• If two propositions can be linked with “If…, then…”,
then we have an implication.
• p is the antecedent and q is the consequent
• The symbol would be 
• For example:
▫ P: You steal
▫ Q: you go to prison
Therefore, the words “If” and “then” are added.
“if you steal, then you go to prison.”
Converse, Inverse, and Contrapositive
• Converse: q  p
• Inverse: p  q
• Contrapositive: q  p
▫ For example:
▫ P: It is raining
▫ Q: I will get wet
 Converse: “If it is raining, then I will get wet.”
 Inverse: “It it isn’t raining, then I won’t get wet.”
 Contrapositive: “It I’m not wet, then it isn’t raining.”
Probability
• Probability is the study of the chance of events
happening
success
P(E) 
total
Combined Events
• The probability of event A or event B happening.
P(A∪B) = P(A)+P(B)
• However, this formula only works with A and B
are mutually exclusive (they cannot happen at
the same time)
• If they are not mutually exclusive, use:
P(A∩B)= P(A)+P(B)-P(A∩B)
Sample Space
• There are various ways to illustrate sample spaces:
• Sample space of possible outcomes of tossing a coin.
• Listing
• Sample space = {H,T}
• 2-D Grids
• Tree Diagram
Theoretical Probability
• A measure of the chance of that event occurring
in any trial of the experiment.
• The formula is:
Using Tree Diagrams
Tree Diagrams – Sampling
• Sampling is the process of selecting an object from a large group of
objects and inspecting it, nothing some features
• The object is either put back (sampling with replacement) or put to
one side (sampling without replacement).
Laws of Probability
Type
Definition
Formula
Mutually Exclusive
Events
Events that cannot
happen at the same time
P(A ∩ B) = 0
P(A  B) = P(A) + P(B)
Combined Events (a.k.a
Addition Law)
Events that can happen
at the same time
P(AB) = P(A) + P(B) –
P(A∩B)
Conditional Probability
The probability of an
even A occurring, given
that event B occurred.
P (A | B) = P (A ∩ B)
P (B)
Independent Events
Occurrence of one event
does NOT affect the
occurrence of the other
P(A ∩ B) = P(A) P(B)