We 
start our Discrete Structure class with the
fundamentals of logic . Today, we will reach
the Propositional Logic .
So, first we give that, what do you mean by
propositional logic. What is logic? So, logic
is the study of logic is the study of reasoning;
we write logic is the study of reasoning .
It is specifically concerned about the reasoning
of two statements . So, or whether the reasoning
is correct or not. It is concerned with whether
the reasoning 
is correct. Mainly logicconcerns about the
relationship between two statements, but logic
does not concerned about the correctness of
the statement. Logic concerns the relationship
among the statements particularly whether
it is correct or not correct or not, but not
concerned the correctness 
of a particular statement.
We we take oneexample first .
One simple example we see . See I write all
professors 
drink tea. Second statement I write that anyone
who drinks tea is a scientist . So, the third
statement gives a relationship between 1 and
2 and we can tell if we think that 1 and 2
are correct, then I can write that therefore,
all professors are scientist .
So, we explained this simple example. See
statement - 1, tells all professors drink
tea statement - 2, tells anyone who drinks
tea is a scientist. So, third statement tells
therefore, all professors are scientist. See
the statement 1 and 2 we do notcheck or we
are not concerned whether the statement is
correct or not. But, if if 1 and 2 are correct,
if statement 1 and 2 are correct, then 3 is
correct then statement 3 is 
is correct. So, this means thelogic. So, this
is logic.
So, what is the advantage of logic or why
it is so important or why we will read logic.
So, first we see that thing.
So, why logic is why logic is important? Why,
we will study. Very simple way I can tell
that logic is useful in clarifying 
in clarifying ordinary writing; whatever I
am writing that that can be literature or
that can be some science topic or ouracademiccontent.
So, that must be logical and there must be
some relationbetween thesentences thatI am
writing or the statements I am writing.
If I considered our technical things why logic
is important, then logic methods are used
are used to prove 
the mathematical theorem . In mathematics;
we can tell that in mathematics mainly logic
is used are to prove the mathematical theorems
in computer science it is used to write the
correct program or to prove the correctness
of the program . In computer science 
logic is used to prove the correctness of
the program.
What do you mean by the correctness of the
program see we have to show or we have to
prove that the programming is doing what it
is supposed to do or what is that is to do
the programs do .
So, with this small introduction now westart
thepropositional logic .
First we define the propositions. What do
you mean by propositions? Simply this is a
sentence or statement, but either it is true
or it is false. Since just now what we read
that logic will see the relationship between
the statements or the correctness, but not
the correctness of a particular statement
. So, we define that a sentence or we can
tell a statement that is either true or false
or false, but not both is called a proposition
.
We see some very simple examples so that our
concept about the proposition is clear . I
give somesimple statements or I write those
statements. First I give the only positive
integer 
that divides 7 are 1 and 7 itself . Second
statement I give that Rabindranath Tagore
won Nobel Prize 
for writing Sesher Kabita . Third statement
I write, Rabindranath Tagore won Nobel Prize
for writing or better I tell for translating
Gitanjali in English .
We see some more examples .
First statements I give for every positive
integer n, there is a prime number greater
than n. Fifth sentence I write, earth is the
only planet where there exist life or that
that contains life . I gave you a different
type of sentence, attend all the lecture classes
.
Now with these six sentence I try to explain
that what do we mean by proposition. We see
that first one I told the only positive integer
that divided 7 are 1 and 7 itself. So, it
is true because 7 is a prime number . Then
second statement Rabindranath Tagore won Nobel
Prize for writing Sesher Kabita which is not
correct, which is false . So, now, we we start
writing that our first statement is true,
second statement is false, third statement
Rabindranath Tagore won won Nobel Prize for
translating Gitanjali in English that is true
which is we know that is a true statement
.
Now, next sentence that for every positive
integer n, there is a prime number greater
than n which tells us that the prime numbers
or number of prime numbers are infinite, which
is true. So, this is also true . Fifth sentence
or step fifth statement tells earth is the
only planet that contains life . See it may
be true or false, but at a time either it
is true or false not both. So, I can write
the fifth sentence that may be true or false
till that we know that it is true, but it
may be false. So, it it it is true or false,
but not at the same time it is not it cannot
be true or false.
Sixth sentence I told attend all the lecture
classes. See I cannot tell anything about
the correctness, true or false. This is actually
a command, ok. So, these I cannot tell about
the this is actually a command type of command
type of sentence command type of statements,
that is neither true nor false . So, what
I got that previous three I gotone true statement
- 1 true, statement - 3 true, say statement
- 2 false, statement - 4 true, statement - 5
true or false and statement - 6 I cannot tell
true or false it is actually a different type
of thing it is a ah.
So, I can now conclude or I can write from
these examples what I can tell that statements
statements 1, 2, 5 are propositions are propositions
. But, statement - 6, 6 is not a proposition
not a proposition . So, statement or a declarative
statement which is either true or false, but
not both at a time is a proposition, but the
command or advice this type of sentence or
statements cannot be a proposition. So, we
define proposition with this examples .
Normally we denotepropositions by this; say
p, I can tell that this is a p is proposition
I can write p, q, r and that can be that denotes
one particular statement that which is either
true or false that can be a proposition . So,
normally we call these are also sometimes
we call propositions or primitive statements.
What is primitive statements? That primitive
statements cannot be broken into some simpler
sentence, ok. So, this is primitive statement
or proposition .
Now, wedefine someoperations on this proposition,
ok . See these operations can be either unary
or binary. What is unary; that means, this
operation only operate only one single proposition
. Say we consider p is a p is a proposition,
p is a proposition. Then we operation first
operation we define is a negation, normally
we denote as by this or a write with this
symbol we write negation; that means, if I
write p is a proposition if I write negation
p then it is written that not p not p.
Example; if I give an example say I give an
example. Simple say p is a proposition: it
is raining . First thing is, it is a declarative
statement either true or false. So, what will
be what will be negation p? Negation p is
it is not raining it is not raining . This
is a unary operator ok this is an unary operator
because unary operation. Since it needs only
one proposition or one variable I can take
that one propositional variable p it works
on that .
Now,some complex proposition can be obtained
from the primitive statement or from that
primitive proposition. We define that primitive
statement cannot be broken into simpler statements
or simpler propositions there is a basic thing.
So, some complex proposition how we can get
from there. We have some twooperation for
this .
We that operations are now these are binary
operations. It is conjunctions and disjunctions
conjunction and disjunction . So, these are
two binary operations. So, for this operation
I need two proposition . I take the example
and take quickly the example. p is one proposition:
it is raining, q is: it is cold . Now, conjunction
we define conjunction we defined like that
normally we denote this thing as this symbol
we write p and q and it tells us that it is
raining and it is cold.
What is disjunction? We denote by this symbol
called p normally called or and this is it
is raining or it is cold . So, these are theare
three operations that negation, AND and OR
these are the basic three operations on the
proposition will be ready. So, we can what
earlier we told that statements and these
are also propositions, these are some compound
propositions obtained from there.
So, they are either true or false since they
are propositions. So, negation p or p and
q or p or q they are either they are also
either true or false, since they are also
proposition .
Now,another we can tell that it isnormally
this or these[conjun/conjunction] conjunction
is and ok conjunction is and conjunction we
can tell this is and. Disjunction is of two
type normally we call or it is of either inclusive
or what is our normal or another is called
the exclusive or when the disjunction is operated
on two at least two proposition, then since
it is binary true proposition then it will
be either inclusive on or it will be exclusive
.
Now, we we we will define hmsome truth table
of these is op with respect to this operation.
So, what is the truth table? Truth table will
give the truth values of the propositions
which is either either true which is either
true 
or false we denote this true by t or one false
by f or 0.
So, that means, that whenever we are say with
the example that we have given that example
p and q; so, if p is true, p is true q is
also true then what will be thetruth value
of p and q. So, that gives you the the one
truth value of itone particulartruth value
of that truth table.
So, then if p q true then p and q is true
. We can write in this way if p is true, q
is false . So, they can give p and q false.
Since it is and since p and q this gives you
actually and. So, now, I can quickly write
that if p false q true then it will give you
that p and q false, p true q false p and q
false. So, in a table if we give the truth
values for all the assignments of the propositions,
that will give us the truth table.
So, now we see thewhat are the truth table
of this three basic operation .
First what we have seen the negation 
negation . 
Negation is denoted as this sign if p and
q are two propositions . So, truth table gives
the truth table presents the truth value of
the proposition obtained for all the assignments;
here assignments 
means either true or false, of the proposition
or primitive proposition in this case p and
q. So, truth table for negation we see truth
table for negation I can write p negation
p; see p can take only two values, either
true or false. So, all the assignments so,
I write I write true then negation p is false.
Then if p is false negation p is true or I
can also write in this way 0 or 1. So, p and
negation p I can write if it is 1 it is 0,
if it is 0 it is 1. So, this is the truth
table.
Now, we see the conjunction and disjunction;
that means, if p and q then I can truth table
for 
of conjunction . I can write p, q and p and
q ok . So, now, p, q we can take four values
it can be both can be true then we and q true
because and so p true q true. So, p and q
true this is true and false since it is ah.
So, the composition or the conjunction of
p and q are false. Then if it is false ok
just if it is false and true then this is
false if it is both false then it is false.
Now I can write truth table of or I can write
truth table of or which is inclusive or 
inclusive or . So, I cannot draw the truth
table p, q p or q both if it is true, if it
is true then it is true. If it is true or
false since it is or so, it can be true if
it is false or true if anyone is true then
it is true only if both are false then only
it will be false. So, this is inclusive or
ok. Now, there is something called exclusive
or. This is 
we normally denote by this and we can write
if it is both are true 
then it is false if both are false then it
is false if anyone is true then only it is
true .
So, I can write that this is my this is my
inclusive or this is my exclusive or . So,
truth table is very importantfor propositional
logic because it gives the truth values of
a complex proposition, for all the truth values
of the basic proposition or the primitive
proposition.
So, what we have read we we did the three
operation; one is unary operation, the negation
the truth table of that, then we read the
conjunction and disjunction to binary operations
and the truth table of the p and q the conjunction
and the truth table of disjunction. Now, there
are two type of disjunction, one is inclusive
OR one is exclusive OR and we can write the
truth table of this thing, ok.
So, now, Iwe we we finish this class with
these truth values and the truth tables and
next class we will do we will see the other
operations.
