Constructing Truth Tables

When we face a complex expression like $(p \land q) \lor (\neg p \implies q)$, we don't try to solve it all at once. We build a Truth Table step-by-step, moving from the inside of the parentheses outward.

1. The Golden Rule: How many rows?

Before you draw your table, you need to know how many rows to create. The formula is: $Rows = 2^n$ Where $n$ is the number of unique variables ($p, q, r$, etc.).

• 2 Variables ($p, q$): $2^2 = 4$ rows.
• 3 Variables ($p, q, r$): $2^3 = 8$ rows.

2. The Step-by-Step Workflow

To solve $(p \lor q) \implies \neg p$, follow this order:

1. Columns for Basics: Start with $p$ and $q$.
2. Negations: Add a column for $\neg p$.
3. Parentheses: Solve the "inner" part $(p \lor q)$.
4. The Final Operator: Combine the parts using the $\implies$ connector.

3. Worked Example: $(p \lor q) \implies \neg p$

4. Visualizing the Logic Tree

We can use a tree diagram to see how the "Final Result" is dependent on the sub-results.

5. Automated Testing in Python

When the tables get too large (like 16 or 32 rows), we use Python to generate them for us.

import itertools

# Define the variables
variables = [True, False]
combinations = list(itertools.product(variables, repeat=2)) # p and q

print("p\tq\tResult")
for p, q in combinations:
    # Our expression: (p or q) -> (not p)
    # Remember: A -> B is (not A or B)
    result = not(p or q) or (not p)
    print(f"{p}\t{q}\t{result}")

Next Step: Now that we can evaluate expressions, we will learn how to identify Tautologies (statements that are always True) and Contradictions (statements that are always False).