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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

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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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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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EMPTY SET OR NULL SETEMPTY SET OR NULL SET

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EMPTY SET OR NULL SETEMPTY SET OR NULL SET

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In set theory the concept of an empty set or null set is very important and interesting. Its definition is as follows:  “a set which contains no elements is called as empty set or null set”, and it is sometimes known as void set or vacuous set. It is usually denoted by

∅

∅; inspired by the letter Ø in the Norwegian and Danish alphabets, and not related to the Greek letter Φ.

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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”

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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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Let p and q be propositions. The conjunction of p and q,
denoted by p U 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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Understand Set,Matrix a little confused 

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Understand Set,Matrix a little confused 

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Rules of Probability
The probability of an event is called P(A), which is a positive number less than or equal to 1.
The total probability for all possible exclusive outcomes should be 1. The joint probability of two different events is denoted P(A,B).

AND Rule of Probability:
•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”.

OR Rule of Probability:
•The probability that either one of 2 different events will occur is the sum of their separate probabilities.

NOT Rule of Probability:
•The chance of an event not happening is 1 minus the chance of it happening.

Combining Rule of Probability:
•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.

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Conditional probability is a measure of the probability of an event occurring, given that another event (by assumption, presumption, assertion or evidence) has already occurred. If the event of interest is A and the event B is known or assumed to have occurred, "the conditional probability of A given B", or "the probability of A under the condition B", is usually written as P(A|B) or occasionally PB(A). This can also be understood as the fraction of probability B that intersects with A:

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Law of Total Probability

For two events A and B associated with a sample space S, the sample space can be divided into a set A ∩ B′, A ∩ B, A′ ∩ B, A′ ∩ B′. This set is said to be mutually disjoint or pairwise disjoint because any pair of sets in it is disjoint. Elements of this set are better known as a partition of sample space.

This can be represented by the Venn diagram as shown below. In cases where the probability of occurrence of one event depends on the occurrence of other events, we use the law of total probability theorem.

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Bernoulli Trials

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.