Why “Exclusive”?

• Exclusivity means: “Only one input is true.”
• This contrasts with inclusive OR, which allows both inputs to be true.

XOR captures mutual exclusivity: true when inputs differ. XNOR captures mutual agreement: true when inputs match.

Why OR works but not AND?

• AND is by definition the opposite of mutual exclusivity by requiring both inputs to be true.
• OR is by definition inclusive—true when either or both inputs are true.

This is why we can make OR more specific through mutual exclusivity, but not AND.

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🔍 Gate-by-Gate Semantics

➕ OR Gate

• A OR B means: → At least one of them is true → Inclusive disjunction: A + B

Test if either input is true

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🚫 NOR Gate

• NOT (A OR B) means: → Both must be false → Complement of OR: (A + B)' or A' · B'

Test if all inputs are false

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✖️ AND Gate

• A AND B means: → Both must be true → Conjunction: A · B

Test if all inputs are true

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🛑 NAND Gate

• NOT (A AND B) means: → At least one must be false → Complement of AND: (A · B)' or A' + B'

Test if either input is false

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🔀 XOR Gate

• A XOR B means: → Inputs must differ → Exclusive disjunction: A ⊕ B

Test if all inputs are different

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🔁 XNOR Gate

• NOT (A XOR B) means: → Inputs must be the same → Logical equivalence: (A ⊕ B)' or A ⊙ B

Test if all inputs are the same

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🧠 Clarifying XOR vs XNOR

• XOR is not just “only one is true”—it’s more precisely:

It affirms difference

• XNOR is not just “not only one is true”—it’s more precisely:

It affirms sameness

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🧪 Boolean Derivations

XOR: Exclusive OR (Difference Check)

• Canonical Form:

  A ⊕ B = (A · B') + (A' · B)

• De Morgan Dual:

  A ⊕ B = ((A' · B)' + (A · B'))'

• Truth Table:

$$ \begin{array}{|c|c|c|} \hline A & B & A ⊕ B \\ \hline 0 & 0 & 0 \\ 0 & 1 & 1 \\ 1 & 0 & 1 \\ 1 & 1 & 0 \\ \hline \end{array} $$

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XNOR: Exclusive NOR (Equality Check)

• Canonical Form:

  A ⊙ B = (A · B) + (A' · B')

• Dual Implication Form:

  A ⊙ B = (A + B') · (A' + B)

• Truth Table:

$$ \begin{array}{|c|c|c|} \hline A & B & A ⊙ B \\ \hline 0 & 0 & 1 \\ 0 & 1 & 0 \\ 1 & 0 & 0 \\ 1 & 1 & 1 \\ \hline \end{array} $$

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✅ Summary Table

Gate	Meaning	Expression
OR	At least one is true	A + B
NOR	Both must be false	(A + B)'
AND	All must be true	A · B
NAND	At least one is false	(A · B)'
XOR	Inputs must differ	A ⊕ B
XNOR	Inputs must be the same	(A ⊕ B)'