Activity 2.3.5 — XOR & XNOR Gates¶

Learning Objectives¶

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

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

Vocabulary¶

Vocabulary (click to expand)

Term	Definition
XOR (Exclusive OR)	A logic gate that produces a HIGH output only when the inputs are different (one HIGH, one LOW).
XNOR (Exclusive NOR)	A logic gate that produces a HIGH output only when the inputs are the same (both HIGH or both LOW).
Parity	A property of binary numbers describing whether the number of 1s is odd or even.
Odd Parity	The condition where the number of 1s in a binary word is odd.
Even Parity	The condition where the number of 1s in a binary word is even.
Inequality Detector	A circuit that produces a HIGH output when inputs are different.

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¶

A	B	Y = A xor B
0	0	0
0	1	1
1	0	1
1	1	0

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):

A ----|
      \_
       )⊕---- Y
     _/
B ----|

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¶

A	B	Y = A xnor B
0	0	1
0	1	0
1	0	0
1	1	1

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)':

A ----|
      \_
       )⊙---- Y
     _/
B ----|

Part 3: XOR Properties and Algebra¶

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

XOR Properties¶

Property	Expression	Description
Identity	A ⊕ 0 = A	XOR with 0 returns A
Inverse	A ⊕ 1 = A'	XOR with 1 inverts A
Idempotent	A ⊕ A = 0	XOR with itself = 0
Inverse Property	A ⊕ A' = 1	XOR with complement = 1
Commutative	A ⊕ B = B ⊕ A	Order doesn't matter
Associative	(A ⊕ B) ⊕ C = A ⊕ (B ⊕ C)	Grouping doesn't matter

XOR Implementation with Basic Gates¶

XOR can be implemented using basic gates:

A ⊕ B = A·B' + A'·B

This translates to: - One AND gate with A and B' (A AND NOT B) - One AND gate with A' and B (NOT A AND B) - One OR gate combining the two AND outputs

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

Actually:
       A----|\
            & |\
       B'---|  )---+---\
       A'---|/      )---+--- Y
       B ----------|/

Key insight: The XOR expression A ⊕ B = A·B' + 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:

A	B	Sum (A ⊕ B)	Carry (A·B)
0	0	0	0
0	1	1	0
1	0	1	0
1	1	0	1

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.

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

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:

A	B	A ⊕ B
0	0	?
0	1	?
1	0	?
1	1	?

Show Solution

A	B	A ⊕ B
0	0	0
0	1	1
1	0	1
1	1	0

XOR = 1 only when inputs are different.

Practice Problem — Simplify XOR Expressions¶

Problem 1: Simplify A ⊕ 0

Problem 2: Simplify A ⊕ A

Problem 3: Simplify A ⊕ A'

Show Solution

Problem 1: A ⊕ 0 = A (Identity property: XOR with 0 returns the other input)

Problem 2: A ⊕ A = 0 (Idempotent property: XOR with itself equals 0)

Problem 3: A ⊕ A' = 1 (Inverse property: XOR with complement equals 1)

Practice Problem — Build XOR with Basic Gates¶

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

Show Solution

The Boolean expression is: A ⊕ B = A·B' + A'·B

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

Steps: 1. Create B' using NOT gate on B 2. Create A' using NOT gate on A 3. AND A with B' → first term (A·B') 4. AND A' with B → second term (A'·B) 5. OR the two terms → final output

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?

Show Solution

a) SUM = A ⊕ B = 1 ⊕ 1 = 0 b) CARRY = A · B = 1 · 1 = 1

Result: 1 + 1 = 10 (binary), meaning SUM = 0 and CARRY = 1

Practice Problem — XNOR Application¶

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

Show Solution

Simply connect A and B to the XNOR gate inputs. The output will be 1 when A = B (both 0 or both 1), indicating equality.

Truth table: - A=0, B=0 → XNOR = 1 (equal) - A=0, B=1 → XNOR = 0 (not equal) - A=1, B=0 → XNOR = 0 (not equal) - A=1, B=1 → XNOR = 1 (equal)

Summary¶

XOR produces HIGH only when inputs are different; XNOR produces HIGH when inputs are the same.
XOR (A ⊕ B) = A·B' + A'·B; XNOR = (XOR)' = A·B + A'·B'
Useful XOR properties: A⊕0=A, A⊕1=A', A⊕A=0, A⊕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·B' + 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