Mathematics
Grade 10
15 min
Truth tables
Truth tables
Tutorial Preview
1
Introduction & Learning Objectives
Learning Objectives
Define a proposition, truth value, and logical connectives (negation, conjunction, disjunction, conditional).
Construct a basic truth table for two propositions, showing all possible truth value combinations.
Construct a truth table for a negation (~p).
Construct truth tables for compound statements involving conjunction (p ∧ q) and disjunction (p ∨ q).
Construct a truth table for a conditional statement (p → q).
Evaluate the truth value of a complex logical statement by building a truth table step-by-step.
Interpret the final column of a truth table to determine when a compound statement is true or false.
If your friend says, "If I finish my homework, then we can play video games," under what single condition is their statement a lie? 🤔
Thi...
2
Key Concepts & Vocabulary
TermDefinitionExample
Proposition (or Statement)A declarative sentence that can be definitively classified as either true or false, but not both."Toronto is the capital of Canada" is a proposition (which is false). "What time is it?" is not a proposition because it is a question.
Truth ValueThe status of a proposition as being either true (T) or false (F).The proposition "The Earth has one moon" has a truth value of True (T).
Negation (~ or ¬)The logical opposite of a proposition. If a proposition 'p' is true, then its negation '~p' is false, and vice versa.If p is "It is raining," then ~p is "It is not raining."
Conjunction (∧)A compound statement that connects two propositions with 'AND'. A conjunction is on...
3
Core Formulas
Conjunction (AND)
p ∧ q is True only when p is True and q is True.
Use this rule to determine the truth value of an 'AND' statement. If even one part is false, the whole statement is false.
Disjunction (OR)
p ∨ q is False only when p is False and q is False.
Use this rule for an 'OR' statement. If you can find at least one true part, the whole statement is true.
Conditional (If...Then)
p → q is False only when p is True and q is False.
This is the rule for an 'if-then' statement. Think of it as a promise; the only way a promise is broken is if the condition is met (p is true) but the result doesn't happen (q is false).
4 more steps in this tutorial
Sign up free to access the complete tutorial with worked examples and practice.
Sign Up Free to ContinueSample Practice Questions
Challenging
Consider the statement '(p ∧ q) → (p ∨ q)'. What is the final column of its truth table?
A.T, F, F, T
B.T, T, F, F
C.T, T, T, T
D.F, F, F, F
Challenging
If the statement '(p ∨ q) → ~r' is False, what must be the truth values of p, q, and r?
A.p is True, q is True, r is True
B.At least one of p or q is True, and r is True
C.p is False, q is False, r is False
D.At least one of p or q is True, and r is False
Challenging
Which statement is a tautology (always true, regardless of the truth values of p and q)?
A.p ∧ ~p
B.p ∨ q
C.p → (p ∧ q)
D.(p ∧ q) → p
Want to practice and check your answers?
Sign up to access all questions with instant feedback, explanations, and progress tracking.
Start Practicing Free