Lesson 2: XOR: exclusive OR and parity (beginner, 10 min)

Exclusive or in three steps: how A XOR B differs from A OR B, why a stairwell light is downstairs XOR upstairs, and why A XOR B XOR C means odd parity rather than “exactly one”. Beginner level, about ten minutes.

Level: beginner10 min11 claims in this lesson were confirmed by the exact checker at build time

Tell inclusive OR from exclusive OR, use XOR for toggles, and see why a chain of XORs means odd parity, not “exactly one”.

The first step sets inclusive and exclusive or side by side on one four-row table; they agree on three rows and part on the row where both inputs are TRUE. The second step makes XOR physical: a light with a switch at the bottom and the top of the stairs, where flipping either switch changes the light whatever the other is set to. The third step is the one people need most, the chain. A XOR B XOR C is TRUE when one or three inputs are TRUE, and the step compares it with the written-out “exactly one of three” to show the row where they differ.

Both comparisons in the lesson are verified claims: the engine re-checks at build time that the pairs differ exactly where the text says. The exercises ask for the value of a three-way XOR when all inputs are TRUE, a truth-table column for the stairwell light, a rewrite of XOR using only AND, OR and NOT, and the assignment on which OR and XOR disagree.

The lesson connects to two pitfall pages (inclusive versus exclusive or, and the parity trap) and to the XOR identities, which prove A XOR A = FALSE and A XOR TRUE = NOT A, the two facts behind toggles and parity bits.

Do this lesson interactively in the app

Steps

1. Inclusive or exclusive?

A OR B is TRUE when at least one input is TRUE. A XOR B is TRUE when the inputs differ: one or the other, but not both.

The two operators agree on three of the four rows and disagree only when both inputs are TRUE.

inclusive := A OR B · outputexclusive := A XOR B · output

Inclusive or exclusive?

#	A	B	inclusive	exclusive
0	FFALSE	FFALSE	FFALSE	FFALSE
1	FFALSE	TTRUE	TTRUE	TTRUE
2	TTRUE	FFALSE	TTRUE	TTRUE
3	TTRUE	TTRUE	TTRUE	FFALSE

Set A and B to TRUE and compare inclusive with exclusive.

not-equivalent inclusive / exclusive · holds

2. XOR as a toggle

A stairwell light with one switch at the bottom and one at the top: light := downstairs XOR upstairs. Flipping either switch changes the light, whatever the other switch shows.

That is the key property of XOR: changing a single input always flips the output.

light := downstairs XOR upstairs · output

downstairs: the switch at the bottom of the stairs is on; upstairs: the switch at the top of the stairs is on; light: the stairwell light is on;

XOR as a toggle

#	downstairs	upstairs	light
0	FFALSE	FFALSE	FFALSE
1	FFALSE	TTRUE	TTRUE
2	TTRUE	FFALSE	TTRUE
3	TTRUE	TTRUE	FFALSE

Start with both switches FALSE, then flip them one at a time and watch light.

3. Chains of XOR: odd parity

A XOR B XOR C is TRUE when an odd number of inputs are TRUE: one or three. It does not mean “exactly one of A, B, C”: with all three TRUE, the chain is TRUE.

“Exactly one” needs its own formula, such as exactly_one here. Compare both outputs on the row where every input is TRUE.

parity := A XOR B XOR C · outputexactly_one := (A AND NOT B AND NOT C) OR (NOT A AND B AND NOT C) OR (NOT A AND NOT B AND C) · output

Chains of XOR: odd parity

#	A	B	C	parity	exactly_one
0	FFALSE	FFALSE	FFALSE	FFALSE	FFALSE
1	FFALSE	FFALSE	TTRUE	TTRUE	TTRUE
2	FFALSE	TTRUE	FFALSE	TTRUE	TTRUE
3	FFALSE	TTRUE	TTRUE	FFALSE	FFALSE
4	TTRUE	FFALSE	FFALSE	TTRUE	TTRUE
5	TTRUE	FFALSE	TTRUE	FFALSE	FFALSE
6	TTRUE	TTRUE	FFALSE	FFALSE	FFALSE
7	TTRUE	TTRUE	TTRUE	TTRUE	FFALSE

Set A, B, and C all to TRUE: parity is TRUE, while exactly_one is FALSE.

not-equivalent parity / exactly_one · holds

Exercises

1. With A=TRUE, B=TRUE, C=TRUE, what is A XOR B XOR C? (predict-output)
2. Fill in the light column for downstairs XOR upstairs (downstairs first). (truth-table)
3. Write a formula equivalent to A XOR B using only AND, OR, and NOT. (write-equivalent)
4. Find the assignment on which A OR B and A XOR B disagree: inclusive is TRUE but exclusive is FALSE. (find-assignment)

Do this lesson interactively in the app

Questions

Is XOR the “or” of everyday speech?

Often, which is the problem: “tea or coffee” is exclusive, while logic's OR is inclusive. The lesson's first step is the table that settles which one a rule needs.

What is XOR with AND, OR and NOT?

(A AND NOT B) OR (NOT A AND B), one term per way the inputs can differ. The third exercise asks for exactly this rewrite, and the XOR elimination law proves it.

Related

Laws

• XOR identities: A XOR FALSE = A, A XOR TRUE = NOT A, A XOR A = FALSE
• XOR elimination: A XOR B = (A AND NOT B) OR (NOT A AND B)
• XNOR elimination: XNOR(A, B) = A ↔ B

Tools

• Truth table generator: any Boolean expression, up to 12 variables, with intermediate columns

See also

• XOR gate: truth table for 2, 3 and 4 inputs (odd parity, not “exactly one”)
• XOR identities: A XOR FALSE = A, A XOR TRUE = NOT A, A XOR A = FALSE
• XOR elimination: A XOR B = (A AND NOT B) OR (NOT A AND B)
• Inclusive vs exclusive or (OR vs XOR), with truth table
• XOR of three inputs is odd parity, not “exactly one”
• Staircase light with three switches: A XOR B XOR C
• Exactly one of three: why XOR is the wrong tool

Lessons

• Lesson 1: NOT, AND, OR: the basic operators (beginner, 10 min)
• Lesson 3: Grouping and precedence (beginner, 12 min)
• Lesson 4: Truth tables (beginner, 12 min)
• Lesson 5: If, only if, unless: necessary and sufficient conditions (intermediate, 15 min)

Generated from the Tangibool engine, semantics tangibool-semantics/1, engine 1.0.0, build 2026-11-07. Every table, map, simplification and verdict on these pages was computed by the Tangibool engine when the site was built, and every claim was re-checked by the same exact checker the app uses. Limits are stated where they apply. © 2026 Foreranger LTD.