Truth tables

✖️ 1. Constructing basic truth tables for single logical operations

🔲 Building Your First Truth Table

• A truth table lists all possible truth values (T or F) for a logical statement.
• For one variable $p$, you need 2 rows (T and F).
• For two variables $p$ and $q$, you need 4 rows (all combinations).
• Basic operations: AND ($p \land q$), OR ($p \lor q$), NOT ($\neg p$).
• AND is true only when both are true; OR is true when at least one is true.

Example: Truth table for $p \land q$:

| $p$ | $q$ | $p \land q$ | |-----|-----|-------------| | T | T | T | | T | F | F | | F | T | F | | F | F | F |

💡 Memory hook: AND needs ALL true; OR needs ONE true.

✖️ 2. Evaluating compound statements using multi-column truth tables

🧩 Breaking Down Complex Statements

• A compound statement combines multiple operations (like $(p \land q) \lor r$).
• Build columns step-by-step from innermost operations outward.
• First evaluate parts inside parentheses, then combine results.
• For three variables, you need 8 rows (double for each new variable).
• Each intermediate step gets its own column before the final answer.

Example: Evaluate $(p \lor q) \land \neg p$ when $p = F$ and $q = T$:

• Step 1: $p \lor q = F \lor T = T$
• Step 2: $\neg p = \neg F = T$
• Step 3: $T \land T = T$

💡 Memory hook: Work inside-out like peeling an onion—innermost operations first.

✖️ 3. Determining logical equivalence of two statements using identical truth columns

⚖️ Testing If Two Statements Are Twins

• Two statements are logically equivalent if their final columns match perfectly.
• Build separate truth tables for each statement with the same variables.
• Compare the final result columns row by row.
• If every row matches, write $\equiv$ between the statements.
• Common equivalence: $\neg(p \land q) \equiv \neg p \lor \neg q$ (De Morgan's Law).

Example: Show $p \land q \equiv q \land p$:

| $p$ | $q$ | $p \land q$ | $q \land p$ | |-----|-----|-------------|-------------| | T | T | T | T | | T | F | F | F | | F | T | F | F | | F | F | F | F |

Final columns match → They are equivalent!

💡 Memory hook: Identical twins have identical DNA—identical columns mean identical logic.

✖️ 4. Applications: Evaluating digital circuit outputs and software condition testing

💻 Real-World Logic Gates and Code

• Digital circuits use AND, OR, NOT gates that follow truth table rules exactly.
• Each gate's output depends on input signals (1 for true, 0 for false).
• Software conditions (if-statements) evaluate using truth tables behind the scenes.
• Testing all input combinations ensures your circuit or code works correctly.
• Truth tables help debug why a condition fails for specific inputs.

Example: Circuit with inputs A and B, output is $(A \land B) \lor \neg A$:

• When $A = 0$ and $B = 1$: $(0 \land 1) \lor 1 = 0 \lor 1 = 1$ (output is HIGH)
• When $A = 1$ and $B = 0$: $(1 \land 0) \lor 0 = 0 \lor 0 = 0$ (output is LOW)

💡 Memory hook: Truth tables are the blueprint—circuits and code are the buildings.