Mathematical Logic: Logical Connectives and their Truth Tables - Discrete Mathematics | Mathematics (2025)

(1) Truth Table for NOT [¬] (Negation) (2) Truth table for AND [ ∧ ] (Conjunction) (3) The truth tables for OR [ ∨ ] (Disjunction)

LogicalConnectives and their Truth Tables

(1) Truth Table forNOT [¬] (Negation)

Truth Table for ¬ p

(2) Truth table forAND [ ∧ ] (Conjunction)

Truth Table for p Λ q

(3) The truth tablesfor OR [ ∨ ] (Disjunction)

Truth Table for p ∨ q

Example 12.12

Writethe statements in words corresponding to ¬ p, p∧ q, p ∨ qand q ∨ ¬p, where p is ‘It is cold’ and qis ‘It is raining.’

Solution

(i) ¬p :It is not cold.

(ii) p ∧ q : It is cold and raining.

(iii) p ∨ q : It is cold or raining.

(iv) q ∨ ¬p : It israining or it is not cold

Observethat the statement formula ¬p has only 1 variable p and its truth table has 2 =(2¹ ) rows. Each of the statement formulae p ∧ q and p ∨ q has two variables p and q . The truth tablecorresponding to each of them has 4 = ( 2² )rows. In general, it follows that if a statement formula involves n variables, then its truth table willcontain 2ⁿ rows.

Example 12.13

How manyrows are needed for following statement formulae?

(i) p ∨ ¬t ∧ ( p∨ ¬s ) (ii) ( ( p∧ q)∨ ( ¬r ∨ ¬s )) ∧ ( ¬t ∧ v)

Solution

(i) ( p ∨ ¬t ) ∧ ( p ∨ ¬s ) contains 3 variables p, s ,and t . Hence the corresponding truthtable will contain 2³ = 8 rows.

(ii) (( p ∧ q)∨ ( ¬ r∨ ¬ s))∧ ( ¬ t∧ v)contains 6 variables p, q, r,s,t, and v . Hence the correspondingtruth table will contain 2⁶ = 64 rows.

ConditionalStatement

Definition 12.13

The conditional statement of any two statements p andq is the statement, “If p , then q ” and it is denoted by p → q . Here p is called the hypothesis orantecedentand q iscalled the conclusion or consequence. p → q is false only if p is true and q is false. Otherwise it is true.

Truth table for p → q

Example 12.14

Considerp → q: If today is Monday, then 4 + 4 = 8.

Here thecomponent statements p and q are given by,

p: Today is Monday; q:4 + 4 = 8.

Thetruth value of p →q is T because the conclusion qis T.

Animportant point is that p →q should not be treated by actuallyconsidering the meanings of p and q in English. Also it is not necessarythat p should be related to q at all.

Consequences

From theconditional statement p →q _, three more conditionalstatements are derived. They are listed below.

(i) Converse statementq → p _.

(ii) Inverse statement¬ p →¬q .

(iii) Contrapositivestatement ¬ q →¬p .

Example 12.15

Writedown the (i) conditional statement (ii) converse statement (iii) inversestatement, and (iv) contrapositive statement for the two statements p and q given below.

p :The number of primes is infinite. q:Ooty is in Kerala.

Solution

Then thefour types of conditional statements corresponding to p and q are respectivelylisted below.

(i) p → q : (conditional statement) “If the number of primes is infinite then Ootyis in Kerala”.

(ii) q → p : (converse statement) “If Ooty is in Kerala then the number of primes is infinite”

(iii) ¬ p →¬q (inverse statement) “If thenumber of primes is not infinite then Ootyis not inKerala”.

(iv) ¬ q →¬p (contrapositive statement) “If Ootyis not inKerala then thenumber of primes is notinfinite”.

Bi-conditionalStatement

Definition 12.14

The bi-conditionalstatement ofany two statements p and q is the statement “ p if and only if q ” and is denoted by p ↔ q .Its truth value is T , whenever both p and q have the same truth values, otherwise it is false.

Truth table for p↔q

Exclusive OR (EOR)[⊽]

Definition 12.15

Let p and q be any two statements. Then p EOR q is such a compound statement that its truth value isdecided by either p or q but not both. It is denoted by p⊽ q . The truth valueof p ⊽ q is T whenever either p orq is T, otherwise it is F. The truth table of p ⊽ q is given below.

Example 12.16

Constructthe truth table for ( p⊽ q)∧ ( p⊽ ¬q).

Also theabove result can be proved without using truth tables. This proof will beprovided after studying the logical equivalence.

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12th Maths : UNIT 12 : Discrete Mathematics : Mathematical Logic: Logical Connectives and their Truth Tables | Discrete Mathematics | Mathematics