Home | | **Maths 12th Std** | Mathematical Logic: Compound Statements, Logical Connectives, and Truth Tables

Any sentence which cannot be split further into two or more statements is called an atomic statement or a simple statement.

**Compound
Statements, Logical Connectives, and Truth Tables**

Any sentence which cannot be split further into two or more
statements is called an **atomic** **statement
**or**
****a
simple statement. **If a statement is
the combination of two or more simple** **statements,
then it is called a **compound statement**
or a **molecular
statement**.
Hence it is clear that any statement can be either a simple statement or a
compound statement.

The
sentences (1), (2), (3) given in example 12.11 are simple statements.

Consider
the statement, “1 is not a prime number and Ooty is in Kerala”.

Note
that the above statement is actually a combination of the following two simple
statements:

*p *: 1 is not a prime number.

*q *: Ooty is in Kerala.

Hence
the given statement is not a simple statement. It is a compound statement.

From the
above discussions, it follows that any simple statement takes the value either *T* or *F*
. So it can be treated as a variable and this variable is known as **statement
variable** or **propositional** **variable**.
The propositional variables are usually denoted by** ***p*,** ***q*,** ***r*** **, ....

To connect two or more simple sentences, we use the words or a
group of words such as “and”, “or”, “if-then”, “if and only if”, and “not”.
These connecting words are known as **logical** **connectives.**

In order to construct a compound statement from simple
statements, some connectives are used**.**
Some basic logical connectives are **negation (not),
conjunction (and)** and **disjunction(or).**

**statement formula**** **is an expression involving one or more statements connected by
some logical connectives.

A table showing the relationship between truth values of simple
statements and the truth values of compound statements formed by using these
simple statements is called **truth table**.

(i) Let *p* be a
simple statement. Then the **negation** of *p* is a statement whose truth value is
opposite to that of *p* . It is denoted
by ¬*p*, read as **not** *p* **.**The truth value of ¬*p* is *T* **,**
if *p* is *F* , otherwise it is *F* .(ii) Let
*p* and *q* be any two simple statements. The **conjunction** of *p* and *q* is obtained by connecting *p* and *q* by the word **and**. It is denoted
by *p* ∧ *q* , read as ‘ *p* conjunction *q* ’ or ‘ *p* hat *q* ’. The truth value of *p* ∧ *q* is *T* , whenever both *p* and *q* are *T* and it is *F* otherwise.

(iii) The **disjunction** of any
two simple statements *p* and *q* is the compound statement obtained by
connecting *p* and *q* by the word ‘or’. It is denoted by *p* ∨ *q* , read as ‘ *p*
disjunction *q* ’ or ‘ *p* cup *q* ’.The truth value of *p* ∨ *q* is *F* , whenever both *p* and *q* are *F* and it is *T* otherwise.

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

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12th Maths : UNIT 12 : Discrete Mathematics : Mathematical Logic: Compound Statements, Logical Connectives, and Truth Tables | Discrete Mathematics | Mathematics

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