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STATEMENTS Statement:   A declarative sentence that is either true or false, but not both is called a statement. Statements a...

Aptitude for Campus Placements - Statements and Prepositional Logic

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.
So all the following declarative sentences are statements.


     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’ (Ù) is called the conjunction of the statement.
          Let p, q are 2 statements, pÙq is a compound statement and its truth table is given by
p
q
pÙq
T
T
T
T
F
F
F
T
F
F
F
F
Ex: p : Today is Sunday
      q : It is raining today.
Then pÙq : Today is Friday and it is raining today.
pÙq is true when both p and q are true and is false otherwise.
Ex:
(i)          p : 2+3=5
       q : 5×8=40
pÙq is true as both p and q are true.
(ii)         p : 5×4=20
       q : 4+5=10
pÙq is false as q is false
(iii)        p : 5+8=12
       q : 5+5=10
pÙq is false as p is false.
(iv)        p : Hyd is capital of M.P.
       q : 18×3=40
pÙq 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 ‘Þ’(if-then) is called conditional or implication.
          If p, q are ‘2’ statements then their implication is denoted by pÞq.
          In this implication p is called hypothesis (or antecedent or premise) and q is called the conclusion (or consequence)
pÞq 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 : x2=4
pÞq : If x=2 then x2=4.
Truth Table:
p
q
pÞq
T
T
T
T
F
F
F
T
T
F
F
F
          The implication pÞq 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
pÞq is true as both p, q are true.
ii)           p : 5×8=40
q : 5+8=12
pÞq is false as q false.
iii)          p : 6÷3=4
q : 7×8=56
pÞq is true as p is false.
iv)          p : 7+8=14
q : 16+9=24
pÞq is true as p and q are false.

Bi-conditional (or) Bi-implication:
 The compound statement of ‘2’ simple statements connected by the connective Û (if and only if) is called Bi-Conditional or Bi-implication.
          If p and q are ‘2’ statements their Bi-conditional is denoted by pÛq.
Ex:     p : ABCD is a parallelogram
          q : AB=CD and BD=AC.
The pÛq : ABCD is a parallelogram if and only if AB=CD, BD=AC. pÛq can also be read as
i)             p is necessary and sufficient for q.
ii)           if p then q and conversely
iii)          p if q. pÛq has exactly the same truth value as (pÞq)Ù(qÞp)
Truth Table:
p
q
pÛq
T
T
T
T
F
F
F
T
F
F
F
T
pÛq is true only when either both p & q are true (or) both are false.
Ex:
i)             p : 4×7=28
       q : 7+4=11
pÛq as both p and q are true
ii)            p : 7×4=28
        q : 7+4=8
pÛq is false as q is false.
iii)          p : 8×4=30
       q : 8+4=12
pÛq is false as p is false.
iv)          p : Chennai is the capital of Karnataka.
        q: Calcutta is in South India.

pÛq is false as both p and q are true.