Logical Operations (Operators)

We can connect statements using various operations. These are called logical operators. With their help, we create new, complex statements.

Negation (¬) – "not"

Negation denies the statement. If p is true, then ¬p is false. If p is false, then ¬p is true.

Example: p = "Today is Monday." → ¬p = "Today is not Monday."

Conjunction (∧) – "and"

Conjunction is true if both p and q are true. If at least one is false, the whole is false.

Example: p = "2 is an even number" (true), q = "5 is greater than 10" (false). → p ∧ q = false.

Disjunction (∨) – "or"

Disjunction is true if at least one statement is true. It is false only if both are false.

Example: p = "3 is less than 5" (true), q = "7 is less than 2" (false). → p ∨ q = true.

Implication (→) – "if…then"

Implication means: if p is true, then q must also be true. It is false only if p is true but q is false.

Example: p = "It is raining" (true), q = "The road is wet" (true). → p → q = true.

Equivalence (↔) – "if and only if"

Equivalence is true if p and q always have the same truth value: both true or both false.

Example: p = "4 is an even number" (true), q = "10 is divisible by 2" (true). → p ↔ q = true.

Summary

The five basic logical operations: ¬ (not), ∧ (and), ∨ (or), → (if…then), ↔ (if and only if). All more complex logical expressions are built from these.

Practice Exercise