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Converse, Contra positive, and Inverse of a Conditional statement:           If p Þ q is a conditional statement then i)            ...

Aptitude for GATE, CRT and PSU exams - Prepositional Logic or Statements

Converse, Contra positive, and Inverse of a Conditional statement:
          If pÞq is a conditional statement then
i)             q Þp is called converse
ii)           ~pÞ~q is called inverse
iii)          ~qÞ~p is called contra positive of pÞq.

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: x2-3x+2=0 " xÎR.
Sol: when x=4, x2-3x+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=4x2
We have to disprove it is odd.
Odd numbers are of the form 2y+1. Where y is an integer.
Hence, by given statement 4x2=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.