Two operations that trip up almost every student in a sets chapter: set difference A - B and symmetric difference A \triangle B. The names rhyme, the symbols are one character apart, and both "remove the overlap" in some sense. A side-by-side picture settles the difference in a glance — one shades a single crescent, the other shades two.

The two definitions

A - B = \{ x \mid x \in A \text{ and } x \notin B \}
A \triangle B = (A - B) \cup (B - A) = \{ x \mid x \in \text{exactly one of } A, B \}

In words: A - B keeps what is in A but not B; A \triangle B keeps what is in exactly one of the two sets, regardless of which one. The second operation is completely symmetric in A and B; the first prefers A.

Why "symmetric": swap A and B in A \triangle B and nothing changes, because "exactly one of the two" does not care which is first. Swap them in A - B and you get a different set, B - A.

The side-by-side picture

Drag the slider to move the right circle B across the rectangle. Two Venn diagrams appear — left shows A - B shaded, right shows A \triangle B shaded. The shaded regions are plainly different whenever the two circles overlap.

Side-by-side Venn diagrams comparing set difference and symmetric differenceTwo Venn diagrams arranged side by side. The left diagram shades only the left crescent representing A minus B. The right diagram shades both crescents representing A symmetric difference B. A slider below controls the overlap. A readout reports the cardinality of each set when A has five elements and B has five with three shared. A − B A B single crescent shaded A △ B A B both crescents shaded drag red dot to change overlap
Left diagram: $A - B$ shades only the left crescent — the part of $A$ that sits outside $B$. Right diagram: $A \triangle B$ shades *both* crescents. The two operations agree only when $B$ is disjoint from $A$ (no overlap to remove). In every other configuration, $A \triangle B$ strictly contains $A - B$, with $B - A$ as the extra piece.

The picture shows the structural fact cleanly: A \triangle B is the union of two set differences.

A \triangle B = (A - B) \cup (B - A)

If you shade only the first term, you get the left diagram. Add the second term and you get the right.

A numerical example

Take A = \{1, 2, 3, 4\} and B = \{3, 4, 5, 6\}.

Operation Result
A - B \{1, 2\}
B - A \{5, 6\}
A \triangle B \{1, 2, 5, 6\}

The symmetric difference has twice the elements of A - B, because it collects both the "A-only" and "B-only" pieces. If you ran the same problem with A = B, then A - B = \varnothing and A \triangle B = \varnothing would agree — but that is a coincidence of the equal-sets case, not a general rule.

Cardinality formulas

Both operations have clean counting rules.

|A - B| = |A| - |A \cap B|
|A \triangle B| = |A| + |B| - 2|A \cap B|

Notice the factor of 2 in the second formula. That is the signature of symmetric difference — you subtract the overlap twice because it appears in both |A| and |B| and must be removed from both crescents.

Why twice: each overlap element gets counted once in |A| and once in |B|. Neither appearance belongs to A \triangle B, so you remove both. Inclusion-exclusion only removes the overlap once because the union includes it; symmetric difference excludes it.

Commutativity — the other big divider

Set difference is not commutative: A - B \neq B - A in general. Drag the slider far enough and you could swap the circle labels, and the left diagram would shade the right crescent instead — a completely different set.

Symmetric difference is commutative: swapping A and B in A \triangle B leaves the shaded region unchanged (both crescents, regardless of which one you call A - B). This symmetry is not a detail — it is the whole reason the operation deserves its own name.

When are they equal?

Only in two special cases.

  1. Disjoint sets. If A \cap B = \varnothing, then B - A is empty, so A \triangle B = A - B \cup \varnothing = A - B = A. The slider shows this: push the circles fully apart and both diagrams shade the same region (the whole of A).
  2. B \subseteq A. If every element of B is already in A, then again B - A = \varnothing and the two operations agree, both equal to A - B.

In all other cases the two operations give different sets, and the size gap is |B - A|.

A worked JEE-style problem

$A = \{x \in \mathbb{N} : x \le 8\}$ and $B = \{x \in \mathbb{N} : 4 \le x \le 10\}$. Find $A - B$, $A \triangle B$, and their cardinalities.

Step 1. Roster the two sets.

A = \{1, 2, 3, 4, 5, 6, 7, 8\}, \quad B = \{4, 5, 6, 7, 8, 9, 10\}

Step 2. Find A - B: keep elements of A not in B.

A - B = \{1, 2, 3\}, \quad |A - B| = 3

Why: walk through A and drop any element that also appears in B. Elements 4 through 8 appear in B, leaving \{1, 2, 3\}.

Step 3. Find B - A.

B - A = \{9, 10\}, \quad |B - A| = 2

Step 4. Union the two to get A \triangle B.

A \triangle B = \{1, 2, 3\} \cup \{9, 10\} = \{1, 2, 3, 9, 10\}, \quad |A \triangle B| = 5

Cardinality check. |A| + |B| - 2|A \cap B| = 8 + 7 - 2 \cdot 5 = 5. Match.

Result. A - B = \{1, 2, 3\} (three elements) and A \triangle B = \{1, 2, 3, 9, 10\} (five elements). The difference is the two extra elements from B - A that the symmetric difference captures but set difference does not.

The two answers differ by exactly |B - A| = 2 elements — which is what the side-by-side Venn picture predicts.

Why the confusion persists

The English vocabulary betrays both operations. "Set difference" suggests subtraction — but subtraction of numbers is not commutative, so students correctly infer that A - B and B - A differ. "Symmetric difference" sounds like "the thing that makes it symmetric" — which it does, but at the cost of adding back the other crescent, not by merely reversing directions.

The picture dissolves both confusions. Whenever a question asks for "everything in A but not B," you want one crescent — A - B. Whenever it asks for "everything that is in exactly one set" or "the disagreement between the two," you want both crescents — A \triangle B.

Related: Set Operations · Symmetric Difference · A − B vs A ∩ B′ · Animated Venn Diagrams