We will use exactly one running example the whole way down:
A={1,2,3,4},B={3,4,5,6}
Read that as: box A holds the labels 1, 2, 3, 4, and box B holds 3, 4, 5, 6. The curly braces { } are just fence-posts around a collection. That's all a set looks like on paper.
PICTURE (s01): on the left, the list — four numbered boxes in a row. On the right, the set — a scattered grid where 1, 2, 3, 4 sit in seemingly random holes. No [0], no [1]. Just presence.
PICTURE (s02): the value 3 enters a funnel labelled hash, out drops a big number, then mod N bends the arrow into bucket #2 of the grid. A second arrow shows 3 entering again later and landing in the same bucket — this repeatability is the whole point.
PICTURE (s03): value 13 and value 3 both compute bucket #2. Under chaining, bucket #2 holds a mini-list [3, 13]; a magnifier shows the equality check newcomer == 3? walking that list. Under open addressing, 13 bounces one slot down to the next empty hole (arrow).
PICTURE (s04): two adds. Adding 5 (empty bucket) → it drops in, green check. Adding 3 again (bucket already holds 3) → red "STOP, already here", the duplicate evaporates. The grid still shows one 3.
PICTURE (s05): left grid (N=8) filling past the 2/3 line (red hatching). An arrow "RESIZE + REHASH" points to a right grid (N=16), where the same values have been recomputed and scattered anew into more, emptier holes.
PICTURE (s06): two circles overlapping. Left crescent = only-A = {1,2} (blue). Lens middle = both = {3,4} (yellow). Right crescent = only-B = {5,6} (red). Outside both, faint "neither: everything else" text. Each element sits in its zone as a labelled dot.
PICTURE (s09): two panels. Left: A - B shades only the left crescent {1,2}. Right: B - A shades only the right crescent {5,6}. A red arrow between them screams "not the same."
Empty B (B=∅): overlap and B-crescent vanish.
A∪∅=A,A∩∅=∅,A−∅=A,A△∅=A
Union/difference/symdiff leave A alone; intersection collapses to nothing (there's nothing to share).
Identical sets (A=B): the circles land perfectly on top of each other — all middle, no crescents.
A∪A=A,A∩A=A,A−A=∅,A△A=∅
Both differences empty out, because "in A but not A" and "in exactly one" are impossible.
Disjoint sets (no shared value): the circles don't overlap — no middle, only crescents.
A∩B=∅,A△B=A∪B
With nothing shared, "exactly one of them" is just "either of them."
PICTURE (s11): three small Venn panels: (a) one circle + a dot-less ghost circle, (b) two circles fully coincident, (c) two circles far apart. Each labelled with the collapsed result.
One Venn diagram, four coloured overlays stacked as a legend: union = all shaded, intersection = lens, difference = left crescent, symmetric difference = both crescents. Below it, the Boolean spine: |→OR→union, &→AND→intersection, -→AND-NOT→difference, ^→XOR→symmetric difference. That single card is the whole derivation.
Recall Feynman: the whole walkthrough in plain words
Picture a wall of empty pigeonholes. To store a number, you run it through a machine (hash) that always sends the same number to the same hole — so before dropping it in you just peek at its hole: if it's already there, you toss the duplicate. That one peek is why sets never repeat and why "is it in here?" is instant — you never walk the wall, you jump straight to the hole.
Two different numbers can point at the same hole — a collision — so the set keeps a tiny list there (or bounces you to the next free hole) and double-checks by actually comparing values. And as the wall fills up, the set quietly builds a bigger wall and re-files everyone (rehashing), so the holes stay mostly empty and lookups stay quick.
Now draw two overlapping bubbles, A and B. Every value falls into one of three visible pockets — only-A, the shared middle, only-B — plus a fourth "outside everything" pocket we never keep. The four operations are just which pockets you keep:
Union (|, OR): keep all three inside pockets — everything in either bubble.
Intersection (&, AND): keep only the middle — what they share.
Difference (A - B): keep only-A — and it's lopsided, B - A keeps the other crescent.
Symmetric difference (^, XOR, written △): keep both crescents, ditch the middle — "in exactly one."
And the corners: an empty bubble makes the middle disappear; two identical bubbles are all middle and no crescents; two far-apart bubbles have no middle at all. Nothing surprises you, because you can always just look at the picture.