Logical Connectives and its Types

The words which are used to combine two or more simple statements are called logical connectives. They are "not", "and", "or", "If... then", and "If and only if".

Conjunction: When two simple statements combine by the word "and" to form a compound statement, such a compound statement is called the conjunction of the original statements. The symbol of the conjunction is ∧.

• Let p ⇒ 4 + 6 = 10 and q ⇒ 2 × 2 = 4. Then the conjunction of the statements p and q is 4 + 6 = 10 and 2 × 2 = 4. It is denoted by p ∧ q and it is true.
• Let p ⇒ 5 + 5 = 10 and q ⇒ 2 × 3 = 4. Then the conjunction of the statements p and q is 5 + 5 = 10 and 2 × 3 = 4. It is denoted by p ∧ q and it is false.

The truth table of the conjunction of the statements p and q is shown below:

Disjunction: When two simple statements combine by the word "or" to form a compound statement, such a compound statement is called the disjunction of the original statements. The symbol of the disjunction is ∨.

Some examples of the disjunctions are:

• Let p: 4 + 6 = 10 and q: 2 × 2 = 4. Then the disjunction of the statements p and q is 4 + 6 = 10 or 2 × 2 = 4. It is denoted by p ∨ q and it is true.
• Let p: 5 + 5 = 10 and q: 2 × 3 = 4. Then the disjunction of the statements p or q is 5 + 5 = 10 or 2 × 3 = 4. It is denoted by p ∨ q and it is true.

The truth table of the disjunction of the statements p and q is shown below:

Negation: A statement which denies the given statement is negation of a given statement. The symbol of the negation is ~.

Some examples of the negation are:

• Let p: "He is smart" is a statement. Then the negation of the statement is ~p: He is not smart.
• Let p: 4 + 5 = 10 (false) be the statement. Then the negation of the statement is ~p: 4 + 5 ≠ 10 (true).

The truth table for the negation of a statement p is shown below:

Conditional or Implication: When two simple statements combine by the word "If ... then" to form a compound statement, such a compound statement is called the conditional of the original statements. The symbol of the conditional is ⇒. It is also called implication.
If p and q are simple statements, then its conditional is symbolized by p ⇒ q. Where, p is known as the antecedent, and q is the consequent.

Note: Antecedent means cause and consequent means result.

Some examples of conditional are:

• Let p: PQR is a triangle and q: the sum of three angles is 180°.
  The conditional of p and q is "If ABC is a triangle then the sum of three angles is 180°." It is symbolized by p ⇒ q.
• Let p: a > b and q: a - b gives a positive result.
  The conditional of p and q is "If a > b then a - b gives a positive result."

The truth table of the conditional of the statements p and q is shown below:

Biconditional or Equivalence: When two simple statements combine by the word "if and only if" to form a compound statement, such a compound statement is called the biconditional of the original statements. The symbol of the conditional is ⇔. It is also called equivalence.
If p and q are simple statements, then their biconditional is symbolized by p ⇔ q. It means p ⇔ q and q ⇔ p.

Some examples of biconditional are:

• Let p: 4 - 3 = 2 and q: 4 > 3.
  The biconditional of p and q is "4 - 3 = 2 if and only if 4 > 3". Where, p ⇔ q is false.
• Let p: I'm smart and q: 2 + 3 = 4.
  The biconditional of p and q is "I'm smart if and only if 2 + 3 = 4." Where, p ⇔ q is true.

The truth table of the conditional of the statements p and q is shown below: