Basic Logical Operations

1. Negation: It means the opposite of the original statement. If p is a statement, then the negation of p is denoted by ~p and read as 'it is not the case that p.' So, if p is true then ~ p is false and vice versa.

Example: If statement p is Paris is in France, then ~ p is 'Paris is not in France'.

p~ p
TF
FT

2. Conjunction: It means Anding of two statements. If p, q are two statements, then "p and q" is a compound statement, denoted by p ∧ q and referred as the conjunction of p and q. The conjunction of p and q is true only when both p and q are true. Otherwise, it is false.

pqp ∧ q
TTT
TFF
FTF
FFF

3. Disjunction: It means Oring of two statements. If p, q are two statements, then "p or q" is a compound statement, denoted by p ∨ q and referred to as the disjunction of p and q. The disjunction of p and q is true whenever at least one of the two statements is true, and it is false only when both p and q are false.

pqp ∨ q
TTT
TFT
FTT
FFF

4. Implication / if-then (⟶): An implication p⟶q is the proposition "if p, then q." It is false if p is true and q is false. The rest cases are true.

pqp ⟶ q
TTT
TFF
FTT
FFF

5. If and Only If (↔): p ↔ q is bi-conditional logical connective which is true when p and q are same, i.e., both are false or both are true.

pqp ↔ q
TTT
TFF
FTF
FFT

Derived Connectors

1. NAND: It means negation after ANDing of two statements. Assume p and q be two propositions. Nanding of pand q to be a proposition which is false when both p and q are true, otherwise true. It is denoted by p ↑ q.

pqp ∨ q
TTF
TFT
FTT
FFT

2. NOR or Joint Denial: It means negation after ORing of two statements. Assume p and q be two propositions. NORing of p and q to be a proposition which is true when both p and q are false, otherwise false. It is denoted by p ↑ q.

pqp ↓ q
TTF
TFF
FTF
FFT

3. XOR: Assume p and q be two propositions. XORing of p and q is true if p is true or q is true but not both and vice-versa. It is denoted by p ⨁ q.

pqp ⨁ q
TTF
TFT
FTT
FFF

Example1: Prove that X ⨁ Y ≅ (X ∧∼Y)∨(∼X∧Y).

Solution: Construct the truth table for both the propositions.

XYX⨁Y∼Y∼XX ∧∼Y∼X∧Y(X ∧∼Y)∨(∼X∧Y)
TTFFFFFF
TFTTFTFT
FTTFTFTT
FFFTTFFF

As the truth table for both the proposition is the same.

Example2: Show that (p ⨁q) ∨(p↓q) is equivalent to p ↑ q.

Solution: Construct the truth table for both the propositions.

pqp⨁q(p↓q)(p⨁q)∨ (p↓q)p ↑ q
TTFFFF
TFTFTT
FTTFTT
FFFTTT