Truth Tables with Two Variables
A free Logic lesson from the “Truth Tables” unit, with a worked example and practice problems including step-by-step solutions.
Two variables create four possible rows: TT, TF, FT, and FF. Listing them systematically prevents missed cases. Learning objective: Build the four-row table for two variables. Prerequisite: No formal prerequisite. Work in this lesson starts with ordinary language, then connects the idea to symbols only after the meaning is clear. Example 1: If p is false, then ¬p is true. Example 2: If p is true and q is false, then p ∧ q is false while p ∨ q is true. A common misconception is to treat familiar wording as proof; instead, check exactly what the statement says and what follows from it.
What you'll learn
- Build the four-row table for two variables
- Explain the idea in plain English before using symbols
- Use examples, non-examples, or counterexamples to check the reasoning
Worked example
Problem. Example case A (Truth Tables with Two Variables): Worked example: Which rule or habit best matches Truth Tables with Two Variables?
- Truth Tables with Two Variables targets a specific reasoning habit.
- Truth Tables with Two Variables focuses on build the four-row table for two variables.
- The other choices either overclaim or change the logical relationship.
Answer: Truth Tables with Two Variables focuses on build the four-row table for two variables.
Practice problems
1. Practice case A (Truth Tables with Two Variables): A truth table with three variables has how many rows?
Show solution
- Warm-up: First identify exactly what the question is asking: Practice case A (Truth Tables with Two Variables): A truth table with three variables has how many rows?
- For symbolic logic, translate the symbols into plain English, then apply the truth condition for the connective.
- Each variable has two truth values.
- Three variables create 2 x 2 x 2 cases.
- That gives 8 rows.
- Check the response against the original logical form: it should match 8 without changing the meaning.
Answer: 8
2. Practice case B (Truth Tables with Two Variables): A truth table with two variables has how many rows?
Show solution
- Warm-up: First identify exactly what the question is asking: Practice case B (Truth Tables with Two Variables): A truth table with two variables has how many rows?
- For symbolic logic, translate the symbols into plain English, then apply the truth condition for the connective.
- Each variable has two truth values.
- Two variables create 2 x 2 cases.
- That gives 4 rows.
- Check the response against the original logical form: it should match 4 without changing the meaning.
Answer: 4
3. Practice case C (Truth Tables with Two Variables): In the row p=True, q=False, r=True, what is p ∨ q?
Choices: True · False
Show solution
- Warm-up: First identify exactly what the question is asking: Practice case C (Truth Tables with Two Variables): In the row p=True, q=False, r=True, what is p ∨ q?
- For symbolic logic, translate the symbols into plain English, then apply the truth condition for the connective.
- p is True and q is False.
- An inclusive-or statement is true when at least one part is true.
- The final value is True.
- Verify the selected choice by checking that it preserves the stated logical meaning and that the other choices change the rule or claim.
Answer: True
4. Practice case D (Truth Tables with Two Variables): In the row p=True, q=False, r=False, what is p → q?
Choices: True · False
Show solution
- Warm-up: First identify exactly what the question is asking: Practice case D (Truth Tables with Two Variables): In the row p=True, q=False, r=False, what is p → q?
- For symbolic logic, translate the symbols into plain English, then apply the truth condition for the connective.
- p is True and q is False.
- A conditional is false only when p is true and q is false.
- The final value is False.
- Verify the selected choice by checking that it preserves the stated logical meaning and that the other choices change the rule or claim.
Answer: False
5. Practice case E (Truth Tables with Two Variables): Lesson focus: Which transfer task would show readiness for Truth Tables with Two Variables?
Choices: Apply build the four-row table for two variables in a new sentence or context. · Repeat the exact worked example without changing the context. · Pick an answer before identifying the claim parts. · Use a rule from another lesson because the words sound close.
Show solution
- A transfer task keeps the same objective but changes the surface context.
- Apply build the four-row table for two variables in a new sentence or context.
- That is a better mastery signal than memorizing a practice prompt.
Answer: Apply build the four-row table for two variables in a new sentence or context.
Practice this interactively with instant feedback and an AI tutor.
Practice Truth Tables with Two Variables Take the free placement check