Question: How to use conditional probobility
Answer: Dear student In Conditional probability we put some
condition on an event to be occur. e.g. A pair of dice is
tossed. Find the probability that one of the dice is 2 if
the sum is 6. If we have to find the probability that
one of the dice is 2, then it is the case of simple
probability. Here we put a condition that sum is six.
Now A = { 2 appears in atleast one die} E = {sum is 6 }
Here E = { (1,5), (2, 4), (3, 3), (4, 2), (5, 1) } Here two
order pairs ( 2, 4 ) and ( 4, 2) satisfies the A. (i.e.
belongs to A) Now A (intersection) B= { (2,4), (4,2)}
Now by formula P(A/E) = P(A (intersection) E)/ P(E) =
2/5
Question: In which condition we use combination and in which
condition permutation.
Answer: This depends on the statement of question. If in the
statement of question you finds out that repetition of
objects are not allowed and order matters then we use
Permutation. e.g. Find the number of ways that a party
of seven persons can arrange themselves in a row of
seven chairs. See in this question repetition is no
allowed because whenever a person is chosen for a
particular seat r then he cannot be chosen again and
also order matters in the arrangements of chairs so we
use permutation here. If in the question repetition of
samples are not allowed and order does not matters
then we use combination. A student is to answer eight
out of ten questions on an exam. Find the number m of
ways that the student can choose the eight questions
See in this question repetition is not allowed that is
when you choose one question then you cannot choose
it again and also order does not matters(i.e either he
solved Q1 first or Q2 first) so you use combination in
this question.
Question: What is the differnce between edge and vertex
Answer: Vertices are nodes or points and edges are lines/arcs
which are used to connect the vertices. e.g If you are
making the graph to find the shortest path or for nay
purpose of cites and roads between them which
contain Lahore Islamabad Faisalabad Karachi
them are edges.
Question: What is the differnce between yes and allowed in
graphs.
Answer: Allowed mean that specific property can be occurs in
that case but yes mean that specific property always
occurs in that case. e.g. In Walk you may start and end
at same point and may not be (allowed). But in Closed
Walk you have to start and end at same point (yes).
Question: what is the meanging of induction? and also
Answer: Basic meaning of induction is: a)The act or an instance
of inducting. b) A ceremony or formal act by which a
person is inducted, as into office or military service.
In Mathematics. A two-part method of proving a
theorem involving a positive integral variable. First the
theorem is verified for the smallest admissible value
of the integer. Then it is proven that if the theorem is
true for any value of the integer, it is true for the
next greater value. The final proof contains the two
parts. As you have studied. It also means that
presentation of material, such as facts or evidence, in
support of an argument or a proposition. Whether in
Physics Induction means the creation of a voltage or
current in a material by means of electric or magnetic
fields, as in the secondary winding of a transformer
when exposed to the changing magnetic field caused by
an alternating current in the primary winding. In
Biochemistry,it means that the process of initiating or
increasing the production of an enzyme or other
protein at the level of genetic transcription. In
embryology,it means that the change in form or shape
caused by the action of one tissue of an embryo on
adjacent tissues or parts, as by the diffusion of
hormones or chemicals.
Question: What is "Hypothetical Syllogism".
Answer: Hypothetical syllogism is a law that if the argument is
of the form p --> q q---> r Therefore p---> r Then it'll
always be a tautology. i.e. if the p implies q and q
implies r is true then its conclusion p implies r is always
true.
Question: A set is define a well define collection of distinct
objects so why an empty set is called a set although it
has no element?
Answer: Some time we have collection of zero objects and we
call them empty sets. e.g. Set of natural numbers
greater than 5 and less than 5. A = { x belongs to N /
5< x < 5 } Now see this is a set which have collection of
elements which are greater than 5 and less than 5
( from natural number).
Question: What is improper subset.
Answer: Let A and B be sets. A is proper subset of B, if, and
only if, every element of A is in B but there is at least
on element if B that is not in A. Now A is improper
subset of B, if and only if, every element of A is in B
and there is no element in B which is not in A. e.g. A=
{ 1, 2 , 3, 4} B= { 2, 1, 4, 3} Now A is improper subset of
B. Because every element of A is in B and there is no
element in B which is not in A
Question: FAQ's in document Form
Answer: .
Question: How to check validity and unvalidity of argument
through diagram.
Answer: To check an argument is valid or not you can also use
Venn diagram. We identify some sets from the
premises . Then represent those sets in the form of
diagram. If diagram satisfies the conclusion then it is
a valid argument otherwise invalid. e.g. If we have
three premises S1: all my friends are musicians S2:
John is my friend. S3: None of my neighbor are
musicians. conclusion John is not my neighbor. Now we
have three sets Friends, Musicians, neighbors. Now you
see from premises 1 and 2 that friends are subset of
musicians .From premises 3 see that neighbor is an
individual set that is disjoint from set musicians. Now
represent then in form of Venn diagram. Musicians
neighbour Friends Now see that john lies in set friends
which is disjoint from set neighbors. So their
intersection is empty.Which shows that john is not his
neighbor. In that way you can check the validity of
arguments
Question: why we used venn digram?
Answer: Venn diagram is a pictorial representation of sets.
Venn diagram can sometime be used to determine
whether or not an argument is valid. Real life problems
can easily be illustrate through Venn diagram if you
first convert them into set form and then in Venn
diagram form. Venn diagram enables students to
organize similarities and differences visually or
graphically. A Venn diagram is an illustration of the
relationships between and among sets, groups of
objects that share something in common.
Question: what is composite relation .
Answer: Let A, B, and C be sets, and let R be relation from A to
B and let S be a relation from B to C. Now by
combining these two relations we can form a relation
from A to C. Now let a belongs to A, b belongs to B,
and c belongs to C. We can write relations R as a R b
and S as b S c. Now by combining R and S we write a (R
0 S) c . This is called composition of Relations holding
the condition that we must have a b belongs to B which
can be write as a R b and b S c (as stated above) . e.g.
Let A= {1,2,3,4}, B={a,b,c,d} , C ={x,y,z} and let
R={ {1,a), (2, d), (3, a), (3, b), (3, d) } and S={ (b, x), (b,
z), (c, y), (d, z)} Now apply that condition which is
stated above (that in the composition R O S only those
order pairs comes which have earlier an element is
common in them e.g. from R we have (3, b) and from S
we have ( b, x) .Now one relation relate 3 to b and
other relates b to x and our composite relation omits
that common and relates directly 3 to x.) I do not
understand your second question send it again. Now R
O S ={(2,z), (3,x), (3,z)}
Question: What are the conditions to confirm functions .
Answer: The first condition for a relation from set X to a set Y
to be a function is 1.For every element x in X, there is
an element y in Y such that (x, y) belongs to F. Which
means that every element in X should relate with
distinct element of Y. e.g if X={ 1,2,3} and Y={x, y} Now
if R={(1,x),(2,y),(1,y),(2,x)} Then R will not be a function
because 3 belongs to X but is does not relates with any
element of Y. so R={(1,x),(2,y),(3,y)} can be called a
function because every element of X is relates with
elements of Y. Second condition is : For all elements x
in X and y and z in Y, if (x, y) belongs to F and (x, z)
belongs to F, then y = z Which means that every
element in X only relates with distinct element of Y.
i.e. R={(1,x),(2,y),(2,x), (3,y)} cannot be called as
function because 2 relates with x and y also.
Question: When a function is onto.
Answer: First you have to know about the concept of function.
Function:It is a rule or a machine from a set X to a set
Y in which each element of set X maps into the unique
element of set Y. Onto Function: Means a function in
which every element of set Y is the image of at least
one element in set X. Or there should be no element
left in set Y which is the image of no element in set X.
If such case does not exist then the function is not
called onto. For example:Let we define a function f :
R----R such that f(x)=x^2 (where ^ shows the symbol
of power i.e. x raise to power 2). Clearly every element
in the second set is the image of atleast one element in
the first set. As for x=1 then f(x)=1^2=1 (1 is the
image of 1 under the rule f) for x=2 then f(x)=2^2=4
(4 is the image of 2 under the rule f) for x=0 then
f(x)=0^2=0 (0 is the image of 0 under the rule f) for
x=-1 then f(x)=(-1)^2=1 (1 is the image of -1 under the
rule f) So it is onto function.
Question: Is Pie an irrational number?
Answer: Pi π is an irrational number as its exact value has an
infinite decimal expansion: Its decimal expansion never
ends and does not repeat.
The numerical value of π truncated to 50 decimal
places is:
3.14159 26535 89793 23846 26433 83279
50288 41971 69399 37510
Question: Difference between sentence and statement.
Answer: A sentence is a statement if it have a truth value
otherwise this sentence is not a statement.By truth
value i mean if i write a sentence "Lahore is capital of
Punjab" Its truth value is "true".Because yes Lahore is
a capital of Punjab. So the above sentence is a
statement. Now if i write a sentence "How are you"
Then you cannot answer in yes or no.So this sentence
is not a statement. Every statement is a sentence but
converse is not true.
Question: What is the truth table?
Answer: Truth table is a table which describe the truth values
of a proposition. or we can say that Truth table display
the complete behaviour of a proposition. There fore
the purpose of truth table is to identify its truth
values. A statement or a proposition in Discrete math
can easily identify its truth value by the truth table.
Truth tables are especially valuable in the
determination of the truth values of propositions
constructed from simpler propositions. The main steps
while making a truth table are "first judge about the
statement that how much symbols(or variables) it
contain. If it has n symbols then total number of
combinations=2 raise to power n. These all the
combinations give the truth value of the statement
from where we can judge that either the truthness of
a statement or proposiotion is true or false. In all the
combinations you have to put values either "F" or "T"
against the variales.But note it that no row can be
repeated. For example "Ali is happy and healthy" we
denote "ali is happy" by p and "ali is healthy" by q so
the above statement contain two variables or symbols.
The total no of combinations are =2 raise to power
2(as n=2) =4 which tell us the truthness of a
statement.
Question: how empty set become a subset of every set.
Answer: If A & B are two sets, A is called a subset of B, if, and
only if, every element of A is also an element of B. Now
we prove that empty set is subset of any other set by
a contra positive statement( of above statement) i.e.
If there is any element in the the set A that is not in
the set B then A is not a subset of B. Now if A={} and
B={1,3,4,5} Then you cannot find an element which is in
A but not in B. So A is subset of B.
Question: What is rational and irrational numbers.
Answer: A number that can be expressed as a fraction p/q
where p and q are integers and q\not=0, is called a
rational number with numerator p and denominator q.
The numbers which cannot be expressed as rational
are called irrational number. Irrational numbers have
decimal expansions that neither terminate nor become
periodic where in rational numbers the decimal
expansion either terminate or become periodic after
some numbers.
Question: what is the difference between graphs and spanning
tree?
Answer: First of all, a graph is a "diagram that exhibits a
relationship, often functional, between two sets of
numbers as a set of points having coordinates
determined by the relationship. Also called plot". Or A
pictorial device, such as a pie chart or bar graph, used
to illustrate quantitative relationships. Also called
chart. And a tree is a connected graph that does not
contain any nontrivial circuit. (i.e., it is circuit-free)
Basically, a graph is a nonempty set of points called
vertices and a set of line segments joining pairs of
vertices called edges. Formally, a graph G consists of
two finite sets: (i) A set V=V(G) of vertices (or points
or nodes) (ii) A set E=E(G) of edges; where each edge
corresponds to a pair of vertices. Whereas, a spanning
tree for a graph G is a subgraph of G that contains
every vertex of G and is a tree. It is not neccesary for
a graph to always be a spanning tree. Graph becomes a
spanning tree if it satisfies all the properties of a
spanning tree.
Question: What is the probability ?
Answer: The definition of probability is : Let S be a finite
sample space such that all the outcomes are equally
likely to occur. The probability of an event E, which is
a subset of S, is P(E) = (the number of outcomes in E)/
(the number of total outcomes in S) P(E) = n (E) / n
( S ) This definition is due to ‘Laplace.’ Thus probability
is a concept which measures numerically the degree of
certainty or uncertainty of the occurrence of an event.
Explaination The basic steps of probability that u have
to remember are as under 1. First list out all possible
out comes. That is called the sample space S For
example when we roll a die the all possible outcomes
are the set S i.e. S = {1,2,3,4,5,6} 2. Secondly we have
to find out all that possible outcomes, in which the
probability is required . For example we are asked to
find the probability of even numbers. First we decide
any name of that event i.e E Now we check all the even
numbers in S which are E = {2,4,6} Remember Event is
always a sub-set of Sample space S. 3. Now we apply
the definition of probability P(E) = (the number of
outcomes in E)/ (the number of total outcomes in S)
P(E) = n (E) / n ( S ) So from above two steps we have n
(E) = 3 and n (S) = 6 then P(E) = 3 / 6 = 1/2 which is
probability of an even number.
Question: what is permutation?
Answer: Permutation comes from the word permute which
means " to change the order of." Basically permutation
means a "complete change." Or the act of altering a
given set of objects in a group. In Mathematics point
of view it means that a ordered arrangement of the
elements of a set (here the order of elements matters
but repetition of the elements is not allowed).
Question: What is a function.
Answer: A function say 'f' is a rule or machine from a set A to
the set B if for every element say a of A, there exist a
unique element say b of set such that b=f(a) Where b
is the image of a under f,and a is the pre-image. Note
it that set A is called the domain of f and Y is called
the codomain of f. As we know that function is a rule
or machine in which we put an input,and we get an
output.Like that a juicer machine.We take some
apples(here apples are input) and we apply a rule or a
function of juicer machine on it,then we get the output
in the form of juice.
Question: What is p implies q.
Answer: p--- >q means to "go from hypothesis to a conclusion"
where p is a hypothesis and q is a conclusion. And note
it that this statement is conditioned because the
"truth ness of statement p is conditioned on the truth
ness of statement q". Now the truth value of p--->q is
false only when p is true and q is false otherwise it will
always true. E.g. consider an implication "if you do your
work on Sunday ,I will give you ten rupees." Here p=you
do your work on Sunday (is the hypothesis) , q=I will
give you ten rupees ( the conclusion or promise). Now
the truth value of p---->q will false only when the
promise is braked. i.e. You do your work on Sunday but
you do not get ten rupees. In all other conditions the
promise is not braked.
Question: What is valid and invalid arguments.
Answer: As "an argument is a list of statements called premises
(or assumptions or hypotheses) which is followed by a
statement called the conclusion. " A valid argument is
one in which the premises entail(or imply) the
conclusion. 1)It cannot have true premises and a false
conclusion. 2)If its premises are true, its conclusion
must be true. 3)If its conclusion is false, it must have
at least one false premise. 4)All of the information in
the conclusion is also in the premises. And an invalid
agrument is one in which the premises do not entail(or
imply) the conclusion. It can have true premises and a
false conclusion. Even if its premises are true, it may
have a false conclusion. Even if its conclusion is false,
it may have true premises. There is information in the
conclusion that is not in the premises. To know them
better,try to solve more and more examples and
exercises.
Question: What is domain and co -domain.
Answer: Domain means "the set of all x-coordinates in a
relation". It is very simple,Let we take a function say f
from the set X to set Y. Then domain means a set
which contain all the elements of the set X. And co
domain means a set which contain all the elements of
the set Y. For example: Let we define a function "f"
from the set X={a,b,c,d} to Y={1,2,3,4}. such that
f(a)=1, f(b)=2, f(c)=3, f(d)=1 Here the domain set is
{a,b,c,d} And the co-domain set is {1,2,3,4} Where as
the image set is {1,2,3}.Because f(a)=1 as 1 is the image
of a under the rule 'f'. f(b)=2 as 2 is the image of b
under the rule 'f'. f(c)=3 as 3 is the image of c under
the rule 'f'. f(d)=1as 1 is the image of d under the rule
'f'. because "image set contains only those elements
which are the images of elements found in set X".
Note it that here f is one -one but not onto,because
there is one element '4' left which is the image of
nothing element under the rule 'f'.
Question: What is the difference between k-sample,k-selection,
k-permutation and k-combination?
Answer: Actually, these all terms are related to the basic
concept of choosing some elements from the given
collection.
For it, two things are important:
1) Order of elements .i.e. which one is first,
which one is second and so on.
2) Repetition of elements
So we can get 4 kinds of selections:
1) The elements have both order and
repetition. ( It is called k-sample )
2) The elements have only order, but no
repetition. ( It is called k-permutation )
3) The elements have only repetition, but no
order. ( It is called k-selection )
4) The elements have no repetition and no
order. ( It is called k-combination )
By ADEEL ABBAS, Bhakkar. AdeelAbbasbk@gmail.com
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