Boolean Logic

Every decision a computer makes — every "if this, then that" — boils down to just TWO values: TRUE or FALSE. This chapter reveals the mathematical LOGIC behind every circuit, every condition, and every digital decision ever made! Welcome to **Boolean Logic** — the language of YES and NO! ✅❌ > [!TIP] > **How to use these notes:** Build Truth Tables for EVERY law and gate yourself — don't just memorise them! Focus especially on **DeMorgan's Theorems**, **Basic Logic Gates**, and **Boolean Law simplification** — these carry the heaviest marks every year! --- ## 3.1 📜 Development of Boolean Logic **Boolean Logic** (also called Boolean Algebra) was developed by the mathematician **George Boole** in the 1850s — a system of logic using only TWO values: **TRUE (1)** and **FALSE (0)**. ```mermaid graph LR BOOLE["👨‍🔬 George Boole\n(1854)\n'An Investigation of\nthe Laws of Thought'"] ALGEBRA["🧮 Boolean Algebra\n(Mathematical logic\nsystem)"] COMPUTERS["💻 Modern Computers\n(Built on TRUE/FALSE\ncircuit logic!)"] BOOLE --> ALGEBRA --> COMPUTERS style BOOLE fill:#9C27B0,color:#fff style COMPUTERS fill:#4CAF50,color:#fff ``` > **Why does this matter for computers?** Remember from Chapter 1 — computers are just millions of switches, each either ON or OFF. Boolean Logic gives us the MATHEMATICAL RULES to combine and simplify these ON/OFF (TRUE/FALSE) values — this is literally how ALL digital circuits and computer decision-making work! | Fact | Detail | | :--- | :--- | | **Developed by** | George Boole | | **Published in** | 1854, in his book "An Investigation of the Laws of Thought" | | **Core Idea** | Logic can be expressed using only two values and mathematical operations | | **Modern Use** | Foundation of digital circuits, programming conditions, database queries | --- ## 3.2 🔢 Binary Valued Quantities A **Binary Valued Quantity** is any variable that can hold ONLY ONE of TWO possible values. ```mermaid graph LR T["✅ TRUE\n1\nHIGH\nYES\nON"] F["❌ FALSE\n0\nLOW\nNO\nOFF"] style T fill:#4CAF50,color:#fff style F fill:#F44336,color:#fff ``` ::: grid ::: card ✅ | TRUE / 1 | Represents an affirmative, ON, or HIGH state | Light switch ON, condition satisfied ::: card ❌ | FALSE / 0 | Represents a negative, OFF, or LOW state | Light switch OFF, condition not satisfied ::: **Real-World Examples of Binary Values:** | Real-World Example | TRUE (1) | FALSE (0) | | :--- | :--- | :--- | | Light Switch | ON | OFF | | Door | Open | Closed | | Exam Result | Pass | Fail | | Login Status | Logged In | Logged Out | > [!IMPORTANT] > **Board Exam Tip** > "What is a Binary Valued Quantity?" — **1-mark** question! > Answer: A variable that can take only ONE of TWO possible values — typically represented as TRUE/FALSE or 1/0. --- ## 3.3 🔧 Logical Operations ### 3.3.1 Logical Function or Compound Statement 📝 A **Logical Function** (or Compound Statement) is an expression formed by combining ONE OR MORE Boolean variables using logical operators, producing a TRUE or FALSE result. ``` Example: Y = A AND B Here, Y is the logical function, A and B are Boolean variables, and AND is the logical operator combining them. ``` > **Analogy:** Think of a logical function like a RECIPE. The ingredients (A, B) go IN, an operation (AND/OR/NOT) is applied, and a single RESULT (TRUE or FALSE) comes OUT! --- ### 3.3.2 Logical Operators 🔧 The THREE fundamental logical operators form the foundation of ALL Boolean logic: ```mermaid graph TD OPS["🔧 LOGICAL OPERATORS"] NOT["🚫 NOT\n(Inverts the value)"] AND["✖️ AND\n(TRUE only if BOTH true)"] OR["➕ OR\n(TRUE if AT LEAST ONE true)"] OPS --> NOT OPS --> AND OPS --> OR style NOT fill:#F44336,color:#fff style AND fill:#2196F3,color:#fff style OR fill:#4CAF50,color:#fff ``` ::: grid ::: card 🚫 | NOT (¬ or ') | Reverses/inverts a single value | NOT TRUE = FALSE ::: card ✖️ | AND (·) | TRUE only when BOTH inputs are TRUE | "I'll go out AND it's sunny" — both conditions must hold ::: card ➕ | OR (+) | TRUE when AT LEAST ONE input is TRUE | "I'll eat pizza OR pasta" — either one satisfies it ::: **Symbols Used in Boolean Expressions:** | Operator | Common Symbols | | :--- | :--- | | **AND** | `·` or `∧` or simply juxtaposition (AB) | | **OR** | `+` or `∨` | | **NOT** | `¬` or `'` (apostrophe) or a bar over the variable (Ā) | > [!IMPORTANT] > **Board Exam Tip** > "Name the three basic logical operators." — **1-mark** question! > Answer: **AND, OR, NOT** --- ### 3.3.3 Evaluation of Boolean Expressions using Truth Table 📊 A **Truth Table** lists ALL possible combinations of input values and the corresponding output result — the ultimate "cheat sheet" for any Boolean expression! **Truth Table for AND:** | A | B | A AND B | | :--- | :--- | :--- | | 0 | 0 | 0 | | 0 | 1 | 0 | | 1 | 0 | 0 | | 1 | 1 | **1** | **Truth Table for OR:** | A | B | A OR B | | :--- | :--- | :--- | | 0 | 0 | 0 | | 0 | 1 | **1** | | 1 | 0 | **1** | | 1 | 1 | **1** | **Truth Table for NOT:** | A | NOT A | | :--- | :--- | | 0 | **1** | | 1 | **0** | > **Memory Trick! 🎵** > **AND** = Strict Boss — needs EVERYONE to say yes. > **OR** = Easy-going friend — just needs ONE yes. > **NOT** = The Opposite Person — always disagrees! **Evaluating a Compound Expression: Y = A·B + C** | A | B | C | A·B | Y = A·B + C | | :--- | :--- | :--- | :--- | :--- | | 0 | 0 | 0 | 0 | 0 | | 0 | 0 | 1 | 0 | 1 | | 0 | 1 | 0 | 0 | 0 | | 0 | 1 | 1 | 0 | 1 | | 1 | 0 | 0 | 0 | 0 | | 1 | 0 | 1 | 0 | 1 | | 1 | 1 | 0 | 1 | 1 | | 1 | 1 | 1 | 1 | 1 | > [!IMPORTANT] > **Board Exam Tip** > "Draw the truth table for Y = A + B'." — **2-mark** question, extremely common! > Answer: Evaluate B' (NOT B) first for each row, then apply OR with A. Total rows = 2ⁿ where n = number of variables (here, 2² = 4 rows). --- ## 3.4 🚪 Basic Logic Gates A **Logic Gate** is a physical electronic circuit that implements a Boolean/logical operation — taking one or more inputs and producing ONE output. ### 3.4.1 Inverter (NOT Gate) 🚫 ```mermaid graph LR IN["A"] -->|"NOT Gate\n—▷o—"| OUT["Ā (NOT A)"] style IN fill:#2196F3,color:#fff style OUT fill:#F44336,color:#fff ``` **Truth Table:** | A | Output (Ā) | | :--- | :--- | | 0 | 1 | | 1 | 0 | > The NOT gate has only ONE input and simply FLIPS/inverts it. --- ### 3.4.2 OR Gate ➕ ```mermaid graph LR A["A"] --> GATE["OR Gate"] B["B"] --> GATE GATE --> OUT["A + B"] style GATE fill:#4CAF50,color:#fff ``` **Truth Table:** | A | B | A + B | | :--- | :--- | :--- | | 0 | 0 | 0 | | 0 | 1 | 1 | | 1 | 0 | 1 | | 1 | 1 | 1 | > **OR Gate Rule:** Output is TRUE if AT LEAST ONE input is TRUE. Only FALSE when BOTH inputs are FALSE. --- ### 3.4.3 AND Gate ✖️ ```mermaid graph LR A["A"] --> GATE["AND Gate"] B["B"] --> GATE GATE --> OUT["A · B"] style GATE fill:#2196F3,color:#fff ``` **Truth Table:** | A | B | A · B | | :--- | :--- | :--- | | 0 | 0 | 0 | | 0 | 1 | 0 | | 1 | 0 | 0 | | 1 | 1 | 1 | > **AND Gate Rule:** Output is TRUE ONLY when BOTH inputs are TRUE. Any single FALSE input makes the whole output FALSE. > [!IMPORTANT] > **Board Exam Tip** > "Draw the logic symbol and truth table for AND, OR, and NOT gates." — **3-mark** question, guaranteed every year! --- ## 3.5 📐 Basic Postulates of Boolean Logic **Postulates** are the fundamental, self-evident RULES that form the foundation of Boolean Algebra — accepted as true without needing proof. ::: grid ::: card 1️⃣ | Closure | Combining two Boolean values with AND/OR always gives another Boolean value | 1 AND 0 = 0 (still a valid Boolean value) ::: card 2️⃣ | Identity Elements | 0 is the identity for OR; 1 is the identity for AND | A + 0 = A, A · 1 = A ::: card 3️⃣ | Commutativity | Order doesn't matter for AND/OR | A+B = B+A ::: card 4️⃣ | Distributivity | AND distributes over OR and vice versa | A·(B+C) = A·B + A·C ::: card 5️⃣ | Complement | Every value A has a complement A' such that A+A'=1 and A·A'=0 | Fundamental for NOT operation ::: --- ## 3.6 🔄 Principle of Duality The **Principle of Duality** states that EVERY valid Boolean expression remains valid if you SWAP: - AND ↔ OR - 0 ↔ 1 ```mermaid graph LR ORIG["Original Expression\nA + 0 = A"] SWAP["Swap AND↔OR, 0↔1"] DUAL["Dual Expression\nA · 1 = A"] ORIG --> SWAP --> DUAL style ORIG fill:#2196F3,color:#fff style DUAL fill:#4CAF50,color:#fff ``` **More Duality Examples:** | Original | Dual (swap AND↔OR, 0↔1) | | :--- | :--- | | A + 1 = 1 | A · 0 = 0 | | A + A = A | A · A = A | | A + A' = 1 | A · A' = 0 | > **Analogy:** Duality is like a MIRROR IMAGE — every rule has a "twin" rule where AND and OR simply swap places, and 0/1 swap places too. If one is TRUE, the mirrored version is AUTOMATICALLY true as well! > [!IMPORTANT] > **Board Exam Tip** > "Write the dual of the expression: A·(B+C)" — **1-mark** question! > Answer: Swap · with +, and + with ·: **A + (B·C)** --- ## 3.7 📜 Basic Theorems of Boolean Algebra/Logic These are the LAWS you'll use CONSTANTLY to simplify Boolean expressions — memorise them well! ### 3.7.1 Properties of 0 and 1 | Law | Expression | | :--- | :--- | | **OR with 0** | A + 0 = A | | **OR with 1** | A + 1 = 1 | | **AND with 0** | A · 0 = 0 | | **AND with 1** | A · 1 = A | --- ### 3.7.2 Indempotence Law **"Combining a variable with ITSELF changes nothing."** | Law | Expression | | :--- | :--- | | OR form | A + A = A | | AND form | A · A = A | --- ### 3.7.3 Involution (Double Negation) **"NOT-NOT A equals A"** — flipping twice returns the original! ``` (A')' = A ``` > **Analogy:** Like saying "It's NOT that I DON'T like pizza" — the double negative means you DO like pizza! Two NOTs cancel out. --- ### 3.7.4 Complementarity Law | Law | Expression | | :--- | :--- | | OR form | A + A' = 1 | | AND form | A · A' = 0 | > A value combined with its OPPOSITE via OR is ALWAYS true; combined via AND is ALWAYS false. --- ### 3.7.5 Commutative Law **"Order doesn't matter"** — just like in regular math! | Law | Expression | | :--- | :--- | | OR form | A + B = B + A | | AND form | A · B = B · A | --- ### 3.7.6 Associative Law **"Grouping doesn't matter"** — how you bracket the terms doesn't change the result. | Law | Expression | | :--- | :--- | | OR form | (A+B)+C = A+(B+C) | | AND form | (A·B)·C = A·(B·C) | --- ### 3.7.7 Distributive Law **"AND/OR distribute over each other"** — similar to multiplication distributing over addition in regular math. | Law | Expression | | :--- | :--- | | AND over OR | A·(B+C) = A·B + A·C | | OR over AND | A+(B·C) = (A+B)·(A+C) | > [!NOTE] > **The SECOND Distributive Law is UNIQUE to Boolean Algebra! 🧠** > In regular math, addition does NOT distribute over multiplication (2+(3×4) ≠ (2+3)×(2+4)). But in Boolean Algebra, BOTH directions work! This is a key difference students often miss. --- ### 3.7.8 Absorption Law **"A larger expression gets 'absorbed' into a simpler one."** | Law | Expression | | :--- | :--- | | Form 1 | A + (A·B) = A | | Form 2 | A · (A+B) = A | > **Analogy:** If you already own a "car" (A), owning "a car AND a specific model of that car" (A·B) doesn't add anything NEW — it's absorbed into just owning "a car" (A)! --- ### 3.7.9 Some Other Rules of Boolean Logic Algebra | Rule | Expression | | :--- | :--- | | **Redundancy Law** | A + A'B = A + B | | **Consensus Theorem** | AB + A'C + BC = AB + A'C | **Complete Boolean Laws — Master Summary Table:** | Law Name | OR Form | AND Form | | :--- | :--- | :--- | | Identity | A+0=A | A·1=A | | Null/Dominant | A+1=1 | A·0=0 | | Idempotent | A+A=A | A·A=A | | Complement | A+A'=1 | A·A'=0 | | Commutative | A+B=B+A | A·B=B·A | | Associative | (A+B)+C=A+(B+C) | (A·B)·C=A·(B·C) | | Distributive | A+(B·C)=(A+B)·(A+C) | A·(B+C)=A·B+A·C | | Absorption | A+(A·B)=A | A·(A+B)=A | > [!IMPORTANT] > **Board Exam Tip** > "Simplify the expression: Y = A + A·B" — **2-mark** question, uses Absorption Law! > Answer: By Absorption Law, A + A·B = **A** --- ## 3.8 🔄 DeMorgan's Theorems **DeMorgan's Theorems** are THE most important, most heavily-tested rules in this entire chapter — they explain how NOT interacts with AND/OR! ### 3.8.1 DeMorgan's First Theorem ``` (A · B)' = A' + B' "The complement of an AND expression equals the OR of the individual complements" ``` **Verification via Truth Table:** | A | B | A·B | (A·B)' | A' | B' | A'+B' | | :--- | :--- | :--- | :--- | :--- | :--- | :--- | | 0 | 0 | 0 | 1 | 1 | 1 | **1** ✅ | | 0 | 1 | 0 | 1 | 1 | 0 | **1** ✅ | | 1 | 0 | 0 | 1 | 0 | 1 | **1** ✅ | | 1 | 1 | 1 | 0 | 0 | 0 | **0** ✅ | > Both (A·B)' and A'+B' produce IDENTICAL results in every row — proving the theorem! --- ### 3.8.2 DeMorgan's Second Theorem ``` (A + B)' = A' · B' "The complement of an OR expression equals the AND of the individual complements" ``` **Verification via Truth Table:** | A | B | A+B | (A+B)' | A' | B' | A'·B' | | :--- | :--- | :--- | :--- | :--- | :--- | :--- | | 0 | 0 | 0 | 1 | 1 | 1 | **1** ✅ | | 0 | 1 | 1 | 0 | 1 | 0 | **0** ✅ | | 1 | 0 | 1 | 0 | 0 | 1 | **0** ✅ | | 1 | 1 | 1 | 0 | 0 | 0 | **0** ✅ | > **Memory Trick! 🎵** **"Break the bar, change the operator!"** When you push a NOT bar INSIDE brackets, it BREAKS apart onto each variable, AND the operator FLIPS (AND↔OR)! ```mermaid graph TD A["(A·B)'"] B["Break the bar,\nFlip AND to OR"] C["A' + B'"] A --> B --> C style A fill:#F44336,color:#fff style C fill:#4CAF50,color:#fff ``` > [!IMPORTANT] > **Board Exam Tip** > "State and verify DeMorgan's First Theorem using a truth table." — **3-mark** question, GUARANTEED every year! > Answer: (A·B)' = A'+B'. Verify by constructing truth tables for both sides and showing they match in every row (as shown above). --- ## 3.9 🚪 More about Logic Gates Beyond the basic 3 gates, these DERIVED gates are built by COMBINING basic gates — and are extremely important in circuit design! ```mermaid graph TD MORE["🚪 MORE LOGIC GATES"] NOR["NOR = NOT(OR)"] NAND["NAND = NOT(AND)"] XOR["XOR = Exclusive OR"] XNOR["XNOR = Exclusive NOR"] MORE --> NOR MORE --> NAND MORE --> XOR MORE --> XNOR style NOR fill:#FF9800,color:#fff style NAND fill:#9C27B0,color:#fff style XOR fill:#F44336,color:#fff style XNOR fill:#2196F3,color:#fff ``` ### 3.9.1 NOR Gate (NOT + OR) **Truth Table:** | A | B | A+B | NOR = (A+B)' | | :--- | :--- | :--- | :--- | | 0 | 0 | 0 | **1** | | 0 | 1 | 1 | 0 | | 1 | 0 | 1 | 0 | | 1 | 1 | 1 | 0 | > NOR = OR gate followed by a NOT — output is the exact OPPOSITE of a regular OR gate! --- ### 3.9.2 NAND Gate (NOT + AND) **Truth Table:** | A | B | A·B | NAND = (A·B)' | | :--- | :--- | :--- | :--- | | 0 | 0 | 0 | **1** | | 0 | 1 | 0 | **1** | | 1 | 0 | 0 | **1** | | 1 | 1 | 1 | 0 | > NAND = AND gate followed by a NOT — output is the exact OPPOSITE of a regular AND gate! > [!NOTE] > **NAND and NOR are "Universal Gates"! 🧠** > Fun fact: EITHER a NAND gate ALONE, or a NOR gate ALONE, can be used to build ANY other logic gate (AND, OR, NOT, everything)! This is why they're called "Universal Gates" — extremely important in real hardware design. --- ### 3.9.3 XOR Gate (Exclusive OR) **Truth Table:** | A | B | A ⊕ B | | :--- | :--- | :--- | | 0 | 0 | 0 | | 0 | 1 | **1** | | 1 | 0 | **1** | | 1 | 1 | 0 | > **XOR Rule:** Output is TRUE only when inputs are DIFFERENT from each other. If BOTH are the same (both 0 or both 1), output is FALSE. > **Analogy:** XOR is like an "Either-Or, but NOT Both" situation — "You can have cake OR ice cream" (not both, and not neither)! --- ### 3.9.4 XNOR Gate (Exclusive NOR) **Truth Table:** | A | B | XNOR = (A⊕B)' | | :--- | :--- | :--- | | 0 | 0 | **1** | | 0 | 1 | 0 | | 1 | 0 | 0 | | 1 | 1 | **1** | > **XNOR Rule:** Output is TRUE when inputs are the SAME (both 0 or both 1) — exactly the OPPOSITE of XOR! **All 7 Logic Gates — Complete Summary Table:** | Gate | Symbol | Output TRUE When... | | :--- | :--- | :--- | | **AND** | · | BOTH inputs are 1 | | **OR** | + | AT LEAST ONE input is 1 | | **NOT** | ' | Input is 0 (inverts) | | **NAND** | (·)' | NOT both 1 (opposite of AND) | | **NOR** | (+)' | NEITHER is 1 (opposite of OR) | | **XOR** | ⊕ | Inputs are DIFFERENT | | **XNOR** | (⊕)' | Inputs are the SAME | > [!IMPORTANT] > **Board Exam Tip** > "Differentiate between XOR and XNOR gate with truth tables." — **2-mark** question, very common! > Answer: **XOR** gives TRUE when inputs DIFFER (0,1 or 1,0). **XNOR** gives TRUE when inputs are the SAME (0,0 or 1,1) — it's the exact complement of XOR. --- ## 3.10 🔌 Logic Circuits A **Logic Circuit** combines MULTIPLE logic gates together to implement a COMPLEX Boolean expression. **Example: Building the circuit for Y = A·B + C** ```mermaid graph LR A["A"] --> AND["AND Gate"] B["B"] --> AND AND --> OR["OR Gate"] C["C"] --> OR OR --> Y["Y = A·B + C"] style AND fill:#2196F3,color:#fff style OR fill:#4CAF50,color:#fff ``` **Steps to Draw a Logic Circuit from a Boolean Expression:** 1. **Identify** all the input variables 2. **Identify** the operators used (AND/OR/NOT) and their ORDER 3. **Draw gates** matching each operator, connecting inputs correctly 4. **Follow operator precedence:** NOT is applied FIRST, then AND, then OR (just like math's BODMAS!) > [!IMPORTANT] > **Board Exam Tip** > "Draw the logic circuit for the Boolean expression Y = (A+B)·C'" — **3-mark** question, common practical question! > Steps: (1) OR gate for A+B, (2) NOT gate for C', (3) AND gate combining the OR output with C' output. --- ## ⚠️ Common Errors and Misconceptions | Mistake | What's Wrong | Correct Understanding | | :--- | :--- | :--- | | ❌ Confusing AND and OR truth tables | Different rules for TRUE output | ✅ AND needs BOTH true; OR needs AT LEAST ONE true | | ❌ Applying DeMorgan's without flipping the operator | Only breaking the bar, not swapping AND/OR | ✅ Must BOTH break the bar AND swap AND↔OR | | ❌ Thinking XOR and OR are the same | XOR excludes the "both true" case | ✅ XOR is FALSE when both inputs are TRUE; OR is TRUE in that case | | ❌ Forgetting Distributive Law works BOTH ways in Boolean Algebra | Assuming it only works like regular math | ✅ BOTH A·(B+C)=AB+AC AND A+(BC)=(A+B)(A+C) are valid | | ❌ Thinking NAND/NOR are just "extra" gates with no special use | Underestimating their importance | ✅ NAND and NOR are UNIVERSAL gates — can build any other gate from them alone | | ❌ Applying Absorption Law incorrectly | Confusing which terms get absorbed | ✅ A+(A·B)=A — the SMALLER standalone term "wins" | --- ## 🔑 Quick Revision — Exam Ready! **Basic Operators:** ``` AND (·) → TRUE only if BOTH true OR (+) → TRUE if AT LEAST ONE true NOT (') → Inverts the value ``` **DeMorgan's Theorems — The Most Important Rule!** ``` (A·B)' = A' + B' (First Theorem) (A+B)' = A' · B' (Second Theorem) "Break the bar, flip the operator!" ``` **All 7 Logic Gates:** | Gate | Rule | | :--- | :--- | | AND | Both TRUE → TRUE | | OR | Any TRUE → TRUE | | NOT | Inverts | | NAND | Opposite of AND | | NOR | Opposite of OR | | XOR | Different → TRUE | | XNOR | Same → TRUE | **Key Laws:** - Identity: A+0=A, A·1=A - Null: A+1=1, A·0=0 - Idempotent: A+A=A, A·A=A - Complement: A+A'=1, A·A'=0 - Absorption: A+(AB)=A **Principle of Duality:** Swap AND↔OR and 0↔1 to get the dual expression. --- ## 🎯 Sample Board Exam Questions ### Q1: Very Short Answer [1 mark each] a) Who developed Boolean Logic? **→ George Boole** b) What are the two values used in Boolean Algebra? **→ TRUE (1) and FALSE (0)** c) Write the dual of A + 0 = A. **→ A · 1 = A** d) State DeMorgan's First Theorem. **→ (A·B)' = A' + B'** e) Which gate gives TRUE output only when both inputs differ? **→ XOR Gate** --- ### Q2: Short Answer [2 marks] **Q: Simplify the Boolean expression: Y = A + A'B** Using the Redundancy Law: A + A'B = A + B **Answer: Y = A + B** --- ### Q3: Short Answer [3 marks] **Q: State and verify DeMorgan's Second Theorem using a truth table.** **Statement:** (A+B)' = A' · B' | A | B | A+B | (A+B)' | A' | B' | A'·B' | | :--- | :--- | :--- | :--- | :--- | :--- | :--- | | 0 | 0 | 0 | 1 | 1 | 1 | 1 | | 0 | 1 | 1 | 0 | 1 | 0 | 0 | | 1 | 0 | 1 | 0 | 0 | 1 | 0 | | 1 | 1 | 1 | 0 | 0 | 0 | 0 | Both columns (A+B)' and A'·B' match exactly, verifying the theorem. --- ### Q4: Truth Table [2 marks] **Q: Draw the truth table for Y = AB' + A'B** | A | B | B' | A' | AB' | A'B | Y=AB'+A'B | | :--- | :--- | :--- | :--- | :--- | :--- | :--- | | 0 | 0 | 1 | 1 | 0 | 0 | 0 | | 0 | 1 | 0 | 1 | 0 | 1 | 1 | | 1 | 0 | 1 | 0 | 1 | 0 | 1 | | 1 | 1 | 0 | 0 | 0 | 0 | 0 | *(Notice: this is exactly the XOR truth table!)* --- ### Q5: Circuit Drawing [3 marks] **Q: Draw the logic circuit for Y = (A + B)·C** Steps: (1) Feed A and B into an OR gate. (2) Take the OR gate's output and C into an AND gate. (3) The AND gate's output is Y. --- ## ✏️ Practice Problems 1. Construct the truth table for the expression Y = (A+B)·(A+C). 2. Simplify: Y = A·(A+B) using the Absorption Law. 3. State DeMorgan's First Theorem and apply it to simplify (A'+B')'. 4. Differentiate between NAND and NOR gates with truth tables. 5. Write the dual of the expression: A·(B+C) = A·B + A·C 6. Draw the logic circuit for Y = A'·B + A·B' 7. Explain why NAND and NOR gates are called "Universal Gates." 8. Simplify the expression Y = AB + A'C + BC using the Consensus Theorem. 9. Construct a truth table comparing XOR and XNOR gates for the same two inputs A and B. 10. A circuit has inputs A, B, C, and the Boolean expression Y = A·B + A·C. Simplify this expression using the Distributive Law and draw the simplified logic circuit.