# 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 |

T | F |

F | T |

**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.

p | q | p ∧ q |

T | T | T |

T | F | F |

F | T | F |

F | F | F |

**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.

p | q | p ∨ q |

T | T | T |

T | F | T |

F | T | T |

F | F | F |

**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.

p | q | p ⟶ q |

T | T | T |

T | F | F |

F | T | T |

F | F | F |

**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.

p | q | p ↔ q |

T | T | T |

T | F | F |

F | T | F |

F | F | T |

## 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.

p | q | p ∨ q |

T | T | F |

T | F | T |

F | T | T |

F | F | T |

**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.

p | q | p ↓ q |

T | T | F |

T | F | F |

F | T | F |

F | F | T |

**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**.

p | q | p ⨁ q |

T | T | F |

T | F | T |

F | T | T |

F | F | F |

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

**Solution:** Construct the truth table for both the propositions.

X | Y | X⨁Y | ∼Y | ∼X | X ∧∼Y | ∼X∧Y | (X ∧∼Y)∨(∼X∧Y) |

T | T | F | F | F | F | F | F |

T | F | T | T | F | T | F | T |

F | T | T | F | T | F | T | T |

F | F | F | T | T | F | F | F |

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.

p | q | p⨁q | (p↓q) | (p⨁q)∨ (p↓q) | p ↑ q |

T | T | F | F | F | F |

T | F | T | F | T | T |

F | T | T | F | T | T |

F | F | F | T | T | T |