If pÞq is a
conditional statement then
i)
q Þp is called converse
ii)
~pÞ~q is called inverse
Example 1 :
Statement : In a ∆ABC if AB=AC the ÐB=ÐC
Conditional: In a ∆ABC, if AB=AC then ÐB=ÐC
Converse: In a ∆ABC, if ÐB=ÐC then
AB=AC
Inverse: In a ∆ABC, if AB≠AC then ÐB≠ÐC
Contra positive: In a ∆ABC, if ÐB≠ÐC then
AB≠AC.
Example 2:
Statement : If it
is raining, then the home team wins.
Converse: if the home team wins, then it is raining.
Inverse: If it is not raining then the home team does
not win.
Contra positive: if the home team does not win, then it is not
raining.
p

q

PÞq

qÞp

pÞ~q

~qÞ~p

T

T

T

T

T

T

T

F

F

T

T

F

F

T

T

F

F

T

F

F

T

T

T

T

\ pÞq =~qÞ~p
qÞp=~pÞ~q
Tautology:
The compound statement that is
always true, no matter what the truth value of the statement that occurs in it,
is called a Tautology.
Ex:
i) p v ~p
P

~P

Pv~P

T

F

T

F

T

T

ii) pÞ(pvq)
p

q

pvq

pÞ(pvq)

T

T

T

T

T

F

T

T

F

T

T

T

F

F

F

T

List of some tautologies:
i)
(pÙq)Þp or pÙqÞq.
ii)
qÞ(p v q)
iii)
(pÙq)Þ(p v q)
iv)
~pÞ(pÞq)
v)
(pÙq)Þ(pÞq)
vi)
~(pÞq)Þp
vii)
~(pÞq)Þ~q
viii)
[pÙ(pÞq)]Þq.
Contradiction:
A
compound Statement that is always false is called a Contradiction.
Ex: pÙ~p
P

~P

PÙ~p

T

F

F

F

T

F

List of
some contradictions:
i)
~(pÙqÞp).
ii)
pÙ(~pÙq).
iii)
Contingency:
A statement that is neither a
tautology nor a contradiction is called a contingency.
i)
~p v q
ii)
[(~p v q ) Ù ( p v
~q)]
iii)
[~(pÙq) v (~p v ~q)]
iv)
~p v (pÙq)
v)
~p v q
vi)
Logical Equivalences:
Compound
Statement that have the same truth values in all possible cases are called
logically equivalent.
Some logical Equivalences:
i)
~(p v q) @ ~p Ù~q
ii)
pÙq @ ~(pÞq)
iii)
pvq @ ~pÞq
iv)
pÞq @ ~pvq
v)
(pÞq) Ù (pÞq) @ pÞ(qÙq)
vi)
pÛq @ (pÙq) v
(~pÙ~q
Dual :
The dual of a compound
statement that contains only the logical operators v, Ù and ~
is the statement obtained by replacing each v by Ù each v
and Ù each Ù by v
each T by F, and each F by T.
Ex:
Dual of the statement
(p v F) Ù (q v T) is (pÙT) v (q
v F)
Open Sentences:
A sentence having one or more
variables is called open sentence if it becomes true or false when we replace
the variable by some specific value.
Ex:
2x+5=9 is an open sentence
Quantifiers:
A Quantifier is a word which
tells “how many? “ values.
There
are two types of Quantifiers.
i)
Universal Quantifier: The quantifier “for all” or “for every” or
“for each” is called the universal quantifier and denoted by ".
ii)
Existential Quantifier: The Quantifier “for some” or “there exist at
least one” or “there is” is called existential quantifier and denoted by $.
Negation of
Compound Statement:
~
( p v q ) = ~p Ù ~q
~ ( p Ù q ) =
~p v ~q
~ ( pÞq) = p Ù~q
~(pÛq) = p Û~q
= ~p Ûq
Proofs:
1)
Direct Proof :
In the method of
direct proof we begin with the given statement ‘p’ and through a logical
sequence of steps we end us with a desired result ‘q’.
2)
Disproof:
In this method of proof, we proceed by assuming that the
statement is false. We arrive at a contradiction. There by we conclude that the
desired result is true. Here we have two methods:
a) Disproof by counter example:
Ex: x^{2}3x+2=0
" xÎR.
Sol:
when x=4, x^{2}3x+2=4≠0.
\ The
given statement is not true for all
b) Disproof by contradiction:
The Square of every even number is odd. An even number can
be written as 2x.
i.e
(2x)^{2}=4x^{2}
We have to disprove it is odd.
Odd
numbers are of the form 2y+1. Where y is an integer.
Hence, by given statement 4x^{2}=2y+1 for some
integer x and y. the left hand side divisible by 2. Whereas the right side is
not divisible by 2.
\ The
given statement is false.