Preposition and logical connectivities in Discrete Mathematics

 Preposition

A preposition is a declarative statement that is either true or false, but not both. Prepositions contain connectives and variables. The variables in the preposition are denoted by capital letters(A, B,....Z).

for example:

• India is a state                     

          returns 'false'

• Maharashtra is a country.    

          returns 'false'

• 12-3=9                                 

          returns 'true'

• 10-4=7                               

          returns  'false'

not prepositions  

• A is greater than 12      
• x+3=6
• go and playout

these statements are not prepositions because the statements do not have either a true value or false values.

Connectives in prepositions

• Disconjunction/OR denoted by '∨' 
• conjunction/AND denoted by '∧'
• Not/Negation denoted by '∾'
• if then denoted by '→'
• if and only if '↔'

Disconjuction/OR

It combines two statements using or.

Disconjunction is denoted by '∨'. the result is true when one of the variables is true.

the truth table for disconjunction:

P	Q	P ∨ Q
True	True	True
True	False	True
False	True	True
False	False	False

there are two types of disconjunction :

1.Exclusive or

It combines two statements using or and it means this statement or the other statement but not both statements. it is denoted by the'⨁' sign.

truth table:

P	Q	P ⨁ Q
True	True	False
True	False	True
False	True	True
False	False	False

2.Inclusive or

It combines two statements using or and it means this statement or the other statement and maybe both statements. It is denoted by '⨀' symbol

truth table:

P	Q	P ⨀ Q
True	True	True
True	False	True
False	True	True
False	False	False

Conjunction

It combines two statements using and, but, while, etc.

conjunction is denoted by the'∧' symbol. The result is true when both variables are true.

the truth table for conjunction:

P	Q	P ∧ Q
True	True	True
True	False	False
False	True	False
False	False	False

For example:

• P: It rains and Q: I will carry an umbrella. The conjunction of these two statements is P∧Q: It rains and I will carry an umbrella.

Negation

 denoted by '∿' . the result is true when the given variable is false and vice versa.

the truth table for negation:

P	∿P
True	False
False	True

for example:

• P: I have a pen  and negation of P is ∿P: I have not a pen
• P: I will attend all lectures negation of P is ∿P: I will not attend all lectures
• P: It will rain today negation of P is ∿P: It will not rain today.

Conditional ( if .....then )/Implication

If P then Q, the statement is false when P is true and Q is false. and in other cases it is true

the truth table for implication:

P	Q	P → Q
True	True	True
True	False	False
False	True	True
False	False	True

Biconditional(if and only if)

the result is true when both the variables are either true or false. if P and Q are true then the result is true and if both are false then also the result is true.

denotes by '*' or '⇔'

The truth table for biconditional:

P	Q	P ⇔ Q
True	True	True
True	False	False
False	True	False
False	False	True

Tautologies

the tautology is a case in which all the output are true in any case of the input.

it is true for all prepositional variables.

for example:

prove this is a tautology :(∿P∨Q)→Q

truth table:

P	Q	∿P	∿P∨Q	(∿P∨Q)→Q
True	True	False	True	True
True	False	False	False	True
False	True	True	True	True
False	False	True	False	True

Contradiction

the contradiction is a case in which all the output are false in any case of the input.

it is false for all prepositional variables.

For example:

show that the preposition is a contradiction:

(P∨Q)∧(∿P∧∿Q)

P	Q	P∨Q	∿P	∿Q	(∿P) ∧ ( ∿B)Q	(P ∨ Q) ∧ [( ∿P) ∧ (∿Q)]
True	True	True	False	False	False	False
True	False	True	False	True	False	False
False	True	True	True	False	False	False
False	False	False	True	True	True	False

Contingency

a contingency is a case in which all the output can be true or false for the given inputs.

a contingency is neither tautology nor contradiction.

for example:

∿Q→(P→Q)

P	Q	∿Q	(P→Q)	∿Q→(P→Q)
True	True	False	True	True
True	False	True	False	False
False	True	False	True	True
False	False	True	False	True

Prepositional equivalence/Logical equivalence

if the two given propositional statements are logically equivalent if they have the same value for each input value or they have biconditional is a tautology(should be true for all variables) 

For example:

determine whether the following two statements are logically equivalent or not.

(P→Q) and (∿P∨Q)

P	Q	(P→Q)	∿P	(∿P∨Q)
True	True	True	False	True
True	False	False	False	False
False	True	True	True	True
False	False	True	True	True

the given propositional statements are logically equivalent as their truth table values for each value are the same.

Testing by 2nd method (Bi-conditionality)

P	Q	(P→Q)	(∿P∨Q)	(P→Q) ⇔ (∿P∨Q)
True	True	True	True	True
True	False	False	False	True
False	True	True	True	True
False	False	True	True	True

here, biconditionality is a tautology hence it is proved by the second method also.

Converse, Inverse, and Contrapositive Proposition

If the preposition P⇒Q is true then

• the converse is Q⇒P 
• the inverse is ~p => ~q
• the contrapositive is  ~q => ~p 

The examples on the converse, inverse, and contrapositive:

1. P: John does hard work 

    Q: he will secure good marks

write converse, inverse, and contrapositive of the above statements.

solution:

P⇒Q: John does hard work then he will secure good marks                           prepositon

Q⇒P: John will secure good marks when he does hard work                          converse

~p => ~q: John doesn't work hard then he will not secure good marks         inverse

 ~q => ~p : John will not secure good marks when he doesn't work hard.   contrapsitive