__MATHEMATICAL LOGIC__**PROPOSITIONAL LOGIC**

**Introduction:**

A proposition is a declarative statement that is either true
or false but not both.

For example, "My name is Sam" is a declarative
statement but "what is your name?" is not a declarative statement.

**Tautology**:

A compound proposition that is always true irrespective of
the truth values of the constituent propositions is called a tautology.

**Contradiction:**

A compound proposition that is always false irrespective of
the truth values of the constituent propositions is called a contradiction.

Proposition that is neither tautology nor a contradiction is
called a contingency.

**Atomic or primary statements:**

Statements without connectives like 'but', 'or', 'and' are
called atomic or primary statements.

**Conjunction**:

Let P and Q be two
statements. Then the statement "P and Q" denoted by P^Q is called the
conjunction of P and Q.

Statement P^Q is true if both P and Q are true

**Disjunction**:

Let P and Q be two statements. Then the statement "P
and Q" denoted by P^Q is called the conjunction of P and Q.

Statement PvQ is true if any one of P or Q is true

Truth Table for Conjunction, Disjunction and Negation

**Conditional**:

Let P and Q be two statements. Then the statement "if P
then Q" denoted by P ->Q is called the Conditional Statement of P and
Q.

Statement P->Q is false if P is true and Q is false else true

Note: P->Q is not same as Q->P

**Biconditional**:

Let P and Q be two statements. Then the statement " P
if and only if Q" denoted by P< ->Q is called the Biconditional
Statement of P and Q.

Statement P<->Q is true if both P and Q are same else
false.

Note: P<->Q is same as Q<->P

Truth Table for Conditional and Biconditional

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