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• Refers to the value of opportunity (Chance) or Possibility (Possibility).
which can be measured in numbers ( 0 to 1 )
• Used in conjunction with many business decision theories such as
• Expected Return
• Investment risk value or return risk value.
from investment (Risk)
• Queuing Theory
• etc.

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A matrix is a rectangular array of numbers. A matrix with m rows and n columns is calledan m × n matrix. The plural of matrix is matrices. A matrix with the same number of rowsas columns is called square. Two matrices are equal if they have the same number of rowsand the same number of columns and the corresponding entries in every position are equal. 

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A matrix is a rectangular array of numbers. A matrix with m rows and n columns is calledan m × n matrix. The plural of matrix is matrices. A matrix with the same number of rowsas columns is called square. Two matrices are equal if they have the same number of rowsand the same number of columns and the corresponding entries in every position are equal. 

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Matrix Addition                                                                    

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Matrix Addition                                                                    

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The sum of two matrices of the same size is obtained by adding elements in the correspondingpositions. Matrices of different sizes cannot be added, because the sum of two matrices is definedonly when both matrices have the same number of rows and the same number of columns.

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Matrix Product (or Matrix Multiplication)

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Matrix Product (or Matrix Multiplication)

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In Figure 1 the colored row of A and the colored column of B are used to compute the element
cij of AB. The product of two matrices is not defined when the number of columns in the first
matrix and the number of rows in the second matrix are not the same.

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A mathematical expression that has an equal sign andlinear expressions

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A mathematical statement that performs functions ofaddition, subtraction, multiplication, and division

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Equations that can be written in the form where a and b are real numbers, with are linear equations. Examples

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A “system” of equations is a set of equations that are related to
one another (have common variables).

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For example, the red point is not a solution to the system, because it
is not on either line.

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•A set is a collection of objects, things or symbolswhich are clearly defined.
•The individual objects in a set are called themembers or elements of the set.
•A set must be properly defined so that we canfind out whether an object is a member of theset.

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1. Listing the elements
The set can be defined by listing all its
elements, separated by commas and enclosed
within braces.

Example:
B = {2, 4, 6, 8, 10}
X = {a, b, c, d, e}

2. Describing the elements
The set can be defined, where possible, by
describing the elements.

Example:
C = {x : x is an integer, x > – 3 }
This is read as: “C is the set of elements x such
that x is an integer greater than –3.”

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•We relate a member and a set using the symbol ∈.
§If an object x is an element of set A, we write x ∈ A. If
an object z is not an element of set A, we write z ∉ A.
•∈ denotes “is an element of” or “is a member of” or
“belongs to”
•∉ denotes “is not an element of” or “is not a member
of” or “does not belong to”

•Example:
If A = {1, 3, 5} then 1 ∈ A and 2 ∉ A

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An infinite set is a set which is not finite. It is not
possible to explicitly list out all the elements of an
infinite set.
Example:

T = {x : x is a triangle}
N is the set of natural numbers
A is the set of fractions

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If every element of a set B is also a member of a set A,
then we say “B is a subset of A”.
We use the symbol ⊂ to mean “is a subset of” and ⊄ to
mean “is not a subset of”

Example:
A = {1, 3, 5}, B = {1, 2, 3, 4, 5}
So, A ⊂ B because every element in A is also in B.
X = {1, 3, 5}, Y = {2, 3, 4, 5, 6}.
X ⊄ Y because 1 is in X but not in Y.

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A universal set is the set of all elements under consideration,
denoted by capital U or sometimes capital E.

Example: Given that U = {5, 6, 7, 8, 9, 10, 11, 12}, list the
elements of the following sets.
a) A = {x : x is a factor of 60}
b) B = {x : x is a prime number}
Solution: The elements of sets A and B can only be selected
from the given universal set U .
a) A = {5, 6, 10, 12}
b) B = {5, 7, 11}

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The intersection of two sets X and Y is the set of
elements that are common to both set X and set
Y. It is denoted by X ∩ Y and is read ‘X
intersection Y’.

Example:
Draw a Venn diagram to represent the
relationship between the sets
X = {1, 2, 5, 6, 7, 9}, Y = {1, 3, 4, 5, 6, 8}
and Z = {3, 5, 6, 7, 8, 10}
Solution: We find that X ∩ Y ∩ Z = {5, 6},
X ∩ Y = {1, 5, 6}, Y ∩ Z = {3, 5, 6, 8} and X ∩ Z = {5, 6, 7}

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Definition
A proposition (p, q, r, ...) is simply a statement (i.e., a
declarative sentence) with a definite meaning, having a truth
value that’s either true (T) or false (F) (never both, neither,
or somewhere in between).

Example
• p: Today is Sunday.
• q: 1+1=9-8.
• r: Bangkok is the capital of Thailand.
• t: I got “A” in Math for IT II.

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Example: Not Proposition
• p: Let’s drink.
• q: a+b.
• r: Take a rest.
• S: A-B=3.
• t: DO IT NOW!
• u: The School of IT

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Let p and q be propositions. The conjunction of p and q,
denoted by p Ù q, is the proposition “p and q.”

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Let p and q be propositions. The disjunction of p and q, denoted
by p v q, is the proposition “p or q.”

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Conditional Proposition or a Material Implication

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Conditional Proposition or a Material Implication

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It looks like something has changed since you first loaded this page. Reloading the page often fixes the problem. Please reload page to continue contributing.

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•Inverter, Or, And, XOR gate symbols, etc.
•Multi-input gates.
•Logic circuits and examples.
•Adders, “half,” “full,” and n-bit.

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•Inverter (logical NOT, Boolean complement).
•AND gate (Boolean
product).
•OR gate (Boolean sum).
•XOR gate (exclusive-OR sum mod 2).

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The linear equations are moderately understandable. 

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The linear equations are moderately understandable. 

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I still don't quite understand

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Still a little confused

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• n(E) = Method number of things or events. only we are interested in
• n(S) = The total number of possible methods associated with an object or event.
What we're interested in is called Numbers of Samples space.
• P(E) or P = probability of event E.

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Probability: what is the chance that a given event will occur
} Probability is expressed in numbers between 0 and 1.
◦ Probability = 0 means the event never happens.
◦ Probability = 1 means it always happens.
} The total probability of all possible events always sums to 1.
} The probability of an event equals the number of times it
happens divided by the number of total opportunities.
} These numbers can be determined by experiment or by
knowledge of the system.

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◦ The probability of 2 independent events both happening is the product of their individual probabilities.

◦ Called the AND rule because “this event happens AND that event
happens”.

◦ Example: What is the probability of rolling a 2 on one die and a 2 on
a second die?

Solution: For each event, the probability is 1/6,
So the probability of both happening is 1/6 x 1/6 = 1/36.

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◦ The probability that either one of 2 different events will occur is the sum
of their separate probabilities.

Example: The chance of rolling either a 2 or a 3 on a die is :

1/6 + 1/6 = 1/3.

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◦ The chance of an event not happening is 1 minus the chance of it
happening.

◦ Example: The chance of not getting a 2 on a die is :

1 - 1/6 = 5/6.

◦ This rule can be very useful. Sometimes complicated problems are
greatly simplified by examining them backwards.

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◦ More complicated situations involve combining the AND and OR rules.
◦ It is very important to keep track of the individuals involved and not
allow them to be confused. This is the source of most people’s
problems with probability.
◦ Example: What is the chance of rolling 2 dice and getting a 2 and a 5?

The trick is, there are 2 ways to accomplish this: a 2 on die A

and a 5 on die B, or a 5 on die A and a 2 on die B.

Each possibility has a 1/36 chance of occurring, and you want

either one or the other of the 2 events,

So the final probability is 1/36 + 1/36 = 2/36 = 1/18.

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Knowing that an event B occurred might affect the probability that
another event A also occurred

Reduce the effective sample space from S to B, then measure “size” of
A relative to its overlap (if any) in B, rather than relative to S
If A and B are two events, the probability of event A when we already
know that event B has occurred is

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An experiment in which a single action, such as flipping a coin,
is repeated identically over and over. The possible results of
the action are classified as "success" or "failure“ ( only two
possible outcomes).
The binomial probability formula is used to find probabilities
for Bernoulli trials.