Operations on propositions are ways to combine or modify propositions using logical rules. The most common operations are:

1. Negation ($\neg$)

Negation flips the truth value of a proposition. If it’s true, it becomes false, and if it’s false, it becomes true.

Examples

• Proposition: "It is raining."
  Negation: "It is not raining."
• Proposition: "$5>2$" (true).
  Negation: "$5\le 2$" (false).

2. Conjunction ($\wedge$)

Conjunction means "and." A compound proposition with $\wedge$ is true only if both parts are true.

Examples

• Proposition 1: "I am hungry."
  Proposition 2: "I am tired."
  Conjunction: "I am hungry and I am tired."
  This is true only if both are true.
• "$3+2=5\wedge 4-1=3$"
  Both "$3+2=5$" and "$4-1=3$" are true, so the whole statement is true.

3. Disjunction ($\vee$)

Disjunction means "or." A compound proposition with $\vee$ is true if at least one part is true.

Examples

• Proposition 1: "I will study."
  Proposition 2: "I will watch TV."
  Disjunction: "I will study or I will watch TV."
  This is true if I do either or both.
• "$3+2=5\vee 4-1=7$"
  "$3+2=5$" is true, "$4-1=7$" is false.
  Since one part is true, the whole statement is true.

4. Implication ($\to$ or $⟹$)

Implication means "if...then." A statement with $\to$ is false only if the first part is true and the second part is false.

Examples

• Proposition: "If it rains, then I will stay inside."
  This is false only if it rains and I don’t stay inside.
• "$3+2=5\to 4-1=3$"
  Since "$3+2=5$" is true and "$4-1=3$" is true, the whole statement is true.

5. Biconditional ($\leftrightarrow$ or $⟺$)

Biconditional means "if and only if." A statement with $\leftrightarrow$ is true when both parts have the same truth value.

Examples

• Proposition: "I will go to the party if and only if my friend goes."
  This is true if both of us go or neither of us goes.
• "$3+2=5\leftrightarrow 4-1=3$"
  Both statements are true, so the biconditional is true.