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

**Logical
Connectives and their Truth Tables**

**Truth Table for **¬** p**

**Truth Table for p **

**Truth Table for p **∨

** **

Write
the statements in words corresponding to ¬** ***p,* *p*
∧ *q*
, *p* ∨ *q*
and *q* ∨ ¬*p,* where *p* is ‘It is cold’ and *q*
is ‘It is raining.’

(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 is
raining or it is not cold

Observe
that the statement formula ¬*p* has only 1 variable *p* and its truth table has 2 =
(
2^{1} ) rows. Each of the statement formulae *p* ∧ *q* and *p* ∨ *q* has two variables *p* and *q* . The truth table
corresponding to each of them has 4 = ( 2^{2} )
rows. In general, it follows that if a statement formula involves *n* variables, then its truth table will
contain 2* ^{n}* rows.

** **

How many
rows are needed for following statement formulae?

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

(i) ( *p* ∨ ¬*t* ) ∧ ( *p* ∨ ¬*s* ) contains 3 variables *p*, *s* ,and *t* . Hence the corresponding truth
table will contain 2^{3} = 8 rows.

(ii) (
( *p* ∧ *q*)
∨ ( ¬ *r*
∨ ¬ *s*
))
∧ ( ¬ *t*
∧ *v*)
contains 6 variables *p*, *q*, *r*,
*s*,*t*
, and *v *. Hence the corresponding
truth table will contain 2^{6} = 64 rows.

** **

The conditional statement of any two statements *p *and*
q *is the statement, “If *p *, then* q *” and it is denoted by *p* → *q* . Here *p* is called the **hypothesis** or
**antecedent**
and *q *is
called 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*

** **

Consider
*p* → *q*
: If today is Monday, then 4 + 4 = 8.

Here the
component statements *p* and *q* are given by,

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

The
truth value of *p* →
*q* is *T* because the conclusion *q*
is *T***.**

An
important point is that *p* →
*q* should not be treated by actually
considering the meanings of *p *and* q *in English. Also it is not necessary
that* p *should be related to* q *at all.

** **

From the
conditional statement *p* →
*q* _{,} three more conditional
statements are derived. They are listed below.

**(i) Converse statement
***q*** **→** ***p*** **_{.}

**(ii) Inverse statement
**¬** ***p*** **→¬*q*** **.

**(iii) Contrapositive
statement **¬** ***q*** **→¬*p*** **.

** **

Write
down the (i) conditional statement (ii) converse statement (iii) inverse
statement, 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.

Then the
four types of conditional statements corresponding to *p* and *q* are respectively
listed below.

(i) *p *→* q *: (conditional statement) “**If*** *the number of primes is infinite* ***then*** *Ooty
is 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** the
number of primes is **not** infinite **then** Ooty
is **not** in
Kerala”.

(iv) ¬ *q* →¬*p* (contrapositive statement) “**If** Ooty
is **not** in
Kerala **then** the
number of primes is **not
**infinite”.

** **

The **bi-conditional
statement** of
any two statements *p* and *q* is the statement “ *p* if and only if *q* ” and is denoted by *p* ↔ *q* ** .**
Its truth value is

**Truth table for p **↔

** **

**Exclusive OR (EOR)[ ****⊽ ****]**

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

** **

Construct
the truth table for ( *p*
⊽ *q*)
∧ ( *p*
⊽ ¬*q*)
.

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

Tags : Discrete Mathematics | Mathematics , 12th Maths : UNIT 12 : Discrete Mathematics

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

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