Disjunction Examples in Math

Start with the recap, study the fully worked examples, then use the practice problems to check your understanding of Disjunction.

This page combines explanation, solved examples, and follow-up practice so you can move from recognition to confident problem-solving in Math.

Concept Recap

A disjunction P \vee Q is a compound statement that is true whenever at least one of its parts is true.

At least one must be true. Logical OR is inclusive — "P or Q or both" — unlike the exclusive everyday "either/or."

Read the full concept explanation →

How to Use These Examples

  • Read the first worked example with the solution open so the structure is clear.
  • Try the practice problems before revealing each solution.
  • Use the related concepts and background knowledge badges if you feel stuck.

What to Focus On

Core idea: P \vee Q is false only when both P and Q are false; it is true in all three other cases (TT, TF, FT).

Common stuck point: Logical OR is inclusive (includes 'both'). 'XOR' is exclusive (one but not both).

Sense of Study hint: Compare with AND by asking: 'Does at least one part need to be true, or do both?' If at least one, you want OR.

Worked Examples

Example 1

easy
Let p: '2 is even' and q: '2 is odd'. Evaluate p \lor q and explain the result.

Solution

  1. 1
    p: '2 is even' — True. q: '2 is odd' — False.
  2. 2
    p \lor q = T \lor F = T.
  3. 3
    In logic, p \lor q (inclusive or) is true when at least one of p, q is true.

Answer

p \lor q = \text{True}
The logical OR is inclusive: it is true whenever at least one operand is true, and false only when both are false. This differs from the everyday exclusive 'or' which excludes the case where both are true.

Example 2

medium
Find all real x satisfying 'x < -1 or x > 3', and express as a union of intervals.

Practice Problems

Try these problems on your own first, then open the solution to compare your method.

Example 1

easy
Evaluate: (a) 'F \lor T', (b) 'F \lor F', (c) 'T \lor T'.

Example 2

medium
Use a truth table to verify De Morgan's Law: \neg(p \lor q) \equiv \neg p \land \neg q.
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Background Knowledge

These ideas may be useful before you work through the harder examples.

logical statement