STATEMENTS
Statement:
A declarative sentence that is
either true or false, but not both is called a statement. Statements are
denoted by p, q, r.
1.
New Delhi is the capital of India.
2.
2+3=5.
3.
5×3=10.
4.
The son sets in the west.
All the following sentences are not statements.
1.
What time is it?
2.
Read this carefully.
3.
X+3=5
4.
X+Y=A.
Sentences
1 and 2 are not propositions because they are not declarative sentences. Sentence
3 and 4 are not propositions because they are neither true nor false, since the
variables in these sentences have not been assigned values.
Truth Value:
The truthiness (or) falsity of
a statement is called its truth value.
If a
statement is true, its truth value is ‘T’
If a
statement is false, its truth value is ‘F’.
Equivalent Statement:
Two statements are said to be
equivalent if their truth values are same. If p & q are two statements
which are equivalent then it denoted by p@q.
Simple statements:
A statement which does not
contain any other statement as its component part is called a simple statement.
Ex: p :
5+4 = 9
q : Samba is good boy.
Compound Statement:
A statement which is composed
of two or more simple statement with a connective is called a compound
statement.
Truth Table:
A truth table displays the
relationships between the truth values of statements.
Negation of a Statement:
The denial of an assertion contained in a
statement is called its negation.
If p is a statement, then its
negation is denoted by ~p.
The negation of a statement is
generally formed by introducing the word “not”
at a proper place in the statement or by prefixing the statement with
the phrase ‘ it is not the case that ‘.
Ex: p :
Today is Tuesday
~p : Today is not Tuesday
P

~P

T

F

F

T

Connectives:
The logical operators that are
used to form new statement from two or more existing statements are called
connectives.
Conjunction:
Two statements p and q are connected with the
connective ‘and’ (Ã™) is called the conjunction of
the statement.
Let p, q are 2 statements, pÃ™q is a
compound statement and its truth table is given by
p

q

pÃ™q

T

T

T

T

F

F

F

T

F

F

F

F

Ex: p :
Today is Sunday
q : It is raining today.
Then pÃ™q :
Today is Friday and it is raining today.
pÃ™q is
true when both p and q are true and is false otherwise.
Ex:
(i)
p : 2+3=5
q : 5×8=40
pÃ™q is
true as both p and q are true.
(ii)
p : 5×4=20
q : 4+5=10
pÃ™q is
false as q is false
(iii)
p : 5+8=12
q : 5+5=10
pÃ™q is
false as p is false.
(iv)
p : Hyd is capital of M.P.
q : 18×3=40
pÃ™q is
false as both p and q are false.
Disjunction:
The compound statement of two
simple statements connected by the connective ‘or’(v) is known as a
disjunction.
If p, q are two statements their disjunction is denoted by
pvq and its truth table is given by.
p

q

pvq

T

T

T

T

F

T

F

T

T

F

F

F

The
disjunction of ‘2’ statements is false when both the statements are false.
Ex:
(i)
p : 2×3=6
q : Hyderabad is capital of A.P.
pvq is
true as both p and q are true.
(ii)
p : Kakinada is head quarters of East
Godavari
q : 2+7=8
pvq is
true as p is true.
(iii)
p : 4×3=7
q : 7×5=35
pvq is
true as q is true.
(iv)
p : 4×3=16
q : 5×18=100
pvq is
false as both p and q are false.
Conditional or Implication:
The compound statement of two
simple statement connected by the connectivity ‘Ãž’(ifthen)
is called conditional or implication.
If p, q are ‘2’ statements then their
implication is denoted by pÃžq.
In this implication p is called hypothesis (or antecedent
or premise) and q is called the conclusion (or consequence)
pÃžq can
also be read as follows
i)
if p, then q
ii)
if p, q
iii) p is sufficient for
q
iv) q if p
v) q when p
vi) q follows from p.
vii) q whenever p.
Ex: p : x=2
q : x^{2}=4
pÃžq : If
x=2 then x^{2}=4.
Truth
Table:
p

q

pÃžq

T

T

T

T

F

F

F

T

T

F

F

F

The implication pÃžq is false only in the case
that p is true, but q is false. It if true when both p and q are true, and p is
false (no matter what truth value q has).
Ex:
i)
p : 4+5=9
q :
6×7=42
pÃžq is
true as both p, q are true.
ii)
p : 5×8=40
q :
5+8=12
pÃžq is
false as q false.
iii)
p : 6÷3=4
q :
7×8=56
pÃžq is
true as p is false.
iv)
p : 7+8=14
q :
16+9=24
pÃžq is
true as p and q are false.
Biconditional (or) Biimplication:
The compound statement of ‘2’ simple
statements connected by the connective Ã› (if and only if) is called
BiConditional or Biimplication.
If p and q are ‘2’ statements their
Biconditional is denoted by pÃ›q.
Ex: p : ABCD is a parallelogram
q : AB=CD and BD=AC.
The pÃ›q :
ABCD is a parallelogram if and only if AB=CD, BD=AC. pÃ›q can
also be read as
i)
p is necessary and sufficient for q.
ii)
if p then q and conversely
iii)
p if q. pÃ›q has
exactly the same truth value as (pÃžq)Ã™(qÃžp)
Truth Table:
p

q

pÃ›q

T

T

T

T

F

F

F

T

F

F

F

T

pÃ›q is
true only when either both p & q are true (or) both are false.
Ex:
i)
p : 4×7=28
q : 7+4=11
pÃ›q as
both p and q are true
ii)
p : 7×4=28
q : 7+4=8
pÃ›q is
false as q is false.
iii)
p : 8×4=30
q : 8+4=12
pÃ›q is
false as p is false.
iv)
p : Chennai is the capital of
Karnataka.
q: Calcutta is in South India.
pÃ›q is
false as both p and q are true.