Activity 2.3.5 — XOR & XNOR Gates

Learning Objectives

By the end of this lesson, students will be able to:

1. Describe the difference between XOR and XNOR gates and identify their unique truth tables.
2. Apply XOR properties to simplify Boolean expressions.
3. Explain how XOR gates are used in binary addition and comparison applications.
4. Build and verify XOR and XNOR circuits using logic gates.

Vocabulary

Part 1: XOR Gate Fundamentals

The XOR (Exclusive OR) gate is a fundamental building block in digital electronics. Unlike the standard OR gate, the XOR gate produces a HIGH output only when the inputs are different.

XOR Truth Table

Key Observation

XOR is HIGH when:

• A = 0 and B = 1 (inputs are DIFFERENT)
• A = 1 and B = 0 (inputs are DIFFERENT)

XOR is LOW when:

• A = 0 and B = 0 (both are the same - both LOW)
• A = 1 and B = 1 (both are the same - both HIGH)

Key insight: XOR is a “difference detector” - it outputs 1 when the inputs differ, and 0 when they’re the same.

XOR Symbol

The XOR gate uses the symbol A ⊕ B (where ⊕ is the XOR operator):

XOR Gate
Output HIGH when inputs are different

Part 2: XNOR Gate Fundamentals

The XNOR (Exclusive NOR) gate is the complement of XOR. It produces a HIGH output when the inputs are the same.

XNOR Truth Table

Key Observation

XNOR is HIGH when:

• A = 0 and B = 0 (both are LOW - SAME)
• A = 1 and B = 1 (both are HIGH - SAME)

XNOR is LOW when:

• A = 0 and B = 1 (inputs are DIFFERENT)
• A = 1 and B = 0 (inputs are DIFFERENT)

Key insight: XNOR is an “equality detector” or “coincidence detector” - it outputs 1 when inputs match.

XNOR Symbol

XNOR is often written as A ⊙ B or (A ⊕ B)‘:

XNOR Gate
Output HIGH when inputs are the same

Part 3: XOR Properties and Algebra

The XOR operation has several useful properties that can simplify circuit design:

XOR Properties

XOR Implementation with Basic Gates

XOR can be implemented using basic gates:

A ⊕ B = A·\overline{B} + \overline{A}·B

This translates to:

• One AND gate with A and \overline{B} (A AND NOT B)
• One AND gate with \overline{A} and B (NOT A AND B)
• One OR gate combining the two AND outputs

      A----|\
           & |\
      \overline{B}---|  )---+--- Y
           |/      |
      B-----------+      (wait, this isn't right)

Actually:
       A----|\
            & |\
       \overline{B}---|  )---+---\
       \overline{A}---|/      )---+--- Y
       B ----------|/

Key insight: The XOR expression A ⊕ B = A·\overline{B} + \overline{A}·B shows it’s the sum of the cases where one input is 1 and the other is 0.

Part 4: Applications of XOR

Binary Addition (Half Adder)

One of the most important applications of XOR is in binary addition. The SUM bit of a half adder is exactly XOR of the two inputs:

The Sum output = A ⊕ B The Carry output = A · B

Comparison / Equality Detection

XOR can be used to compare two bits:

• If A ⊕ B = 0, then A = B
• If A ⊕ B = 1, then A ≠ B

Extending this to multiple bits requires checking if all XOR outputs are 0.

graph LR
    A["4-bit<br/>Value A"] --> XOR1["XOR"]
    A --> XOR2["XOR"]
    A --> XOR3["XOR"]
    A --> XOR4["XOR"]
    B["4-bit<br/>Value B"] --> XOR1
    B --> XOR2
    B --> XOR3
    B --> XOR4
    XOR1 --> NOR["NOR<br/>(4-input)"]
    XOR2 --> NOR
    XOR3 --> NOR
    XOR4 --> NOR
    NOR --> EQ["Equal?<br/>1 = Same<br/>0 = Different"]

    style XOR1 fill:#bbf,stroke:#333
    style XOR2 fill:#bbf,stroke:#333
    style XOR3 fill:#bbf,stroke:#333
    style XOR4 fill:#bbf,stroke:#333
    style NOR fill:#fdb,stroke:#333
    style EQ fill:#bfb,stroke:#333,stroke-width:2px

Parity Checking

Parity is used for error detection:

• Even parity: Number of 1s in the word is even (including parity bit)
• Odd parity: Number of 1s in the word is odd (including parity bit)

An XOR chain can count the number of 1s:

• Even number of 1s → XOR output = 0
• Odd number of 1s → XOR output = 1

graph LR
    D0["Bit 0"] --> XOR1["XOR"]
    D1["Bit 1"] --> XOR1
    XOR1 --> XOR2["XOR"]
    D2["Bit 2"] --> XOR2
    XOR2 --> XOR3["XOR"]
    D3["Bit 3"] --> XOR3
    XOR3 --> P["Parity Bit<br/>0 = Even<br/>1 = Odd"]

    style XOR1 fill:#bbf,stroke:#333
    style XOR2 fill:#bbf,stroke:#333
    style XOR3 fill:#bbf,stroke:#333
    style P fill:#bfb,stroke:#333,stroke-width:2px

7486 Quad XOR IC

The 7486 is a quad 2-input XOR IC containing four independent XOR gates:

• Package: 14-pin DIP
• Four 2-input XOR gates
• Useful for building comparators, adders, and parity generators

Part 5: XNOR Applications

XNOR is essentially an inverted XOR, making it useful for:

Equality Comparison

XNOR outputs 1 when inputs match, making it perfect for checking if two bits are equal:

• A ⊙ B = 1 means A = B

Inverting Selector

XNOR acts as an inverting data selector:

• When select = 0: output = input
• When select = 1: output = input’ (inverted)

Parity (Even)

XNOR can be used for even parity checking:

• Even number of 1s → XNOR output = 1

Key insight: XOR and XNOR are inverses of each other. XNOR = (XOR)’ and XOR = (XNOR)’.

Practice Problem — XOR Truth Table Verification

Problem: Complete the truth table and verify the XOR pattern:

Practice Problem — Simplify XOR Expressions

Problem 1: Simplify A ⊕ 0

Problem 2: Simplify A ⊕ A

Problem 3: Simplify A ⊕ \overline{A}

Practice Problem — Build XOR with Basic Gates

Problem: Draw the gate-level circuit to implement A ⊕ B using AND, OR, and NOT gates.

      A--------|\
               & |\
      B----NOT--|  )---+--- OR ---- Y
               |/      |
      A----NOT--|\
               & |/
      B---------|/

Practice Problem — XOR in Binary Addition

Problem: For a half adder with inputs A=1 and B=1: a) What is the SUM output? b) What is the CARRY output?

Practice Problem — XNOR Application

Problem: How would you use an XNOR gate to detect if two bits A and B are equal?

Summary

• XOR produces HIGH only when inputs are different; XNOR produces HIGH when inputs are the same.
• XOR (A ⊕ B) = A·\overline{B} + \overline{A}·B; XNOR = (XOR)’ = A·B + \overline{A}·\overline{B}
• Useful XOR properties: A⊕0=A, A⊕1=\overline{A}, A⊕A=0, A⊕\overline{A}=1
• XOR is used in half adders (SUM output) and comparison circuits.
• XNOR is used for equality detection and even parity checking.
• The 7486 is a quad 2-input XOR IC.

Key Reminders

• XOR = “difference detector” (1 when different)
• XNOR = “equality detector” (1 when same)
• Remember the XOR Boolean expression: A ⊕ B = A·\overline{B} + \overline{A}·B
• XOR gates are essential for building arithmetic circuits.
• Parity checking uses XOR to detect errors in data transmission.

Custom activity — adapted from PLTW Digital Electronics