Sets, relations and quantifiers

Venn painter — type a set expression such as (A-B)|C' and the matching regions of a two- or three-set Venn diagram are shaded. Click regions (or rows of the truth table beside it) to paint a shape yourself, and the shortest expression the tool knows for it replaces yours. Each region is one row of the truth table for x ∈ A, x ∈ B, x ∈ C, and hovering over either highlights the other. Type a second expression under compare with to check an identity such as De Morgan’s law: the tool says whether the two shade the same regions, and if not, which regions differ. Fill in the counts |A|, |B|, |A∩B|, … and |U| to see inclusion–exclusion at work: the count in every region, the size of the union and of “none of them”, the size of the shaded set, and a warning when counts are impossible.

Relations — up to eight elements on a circle. Click one dot and then another to toggle the arrow between them; click a dot twice for its loop. The tool checks reflexive, symmetric, antisymmetric and transitive, and for each one that fails it names and draws a witness in red: the missing loop, the unreturned arrow, the two-way pair, or the two-step path with no shortcut. The buttons add the reflexive, symmetric or transitive closure. An equivalence relation colours the dots by class and lists the partition; a partial order is redrawn as a Hasse diagram with its incomparable pairs. Relations can also be typed as rules: divides, lt (<), le (≤), eq, ne, subset (⊆ on labels such as ∅,a,b,ab), mod3 (any modN), empty or all, or as arrows 1>2,2>3.

∀∃ game — a statement with two quantifiers over {0, …, n − 1}, such as ∀x ∃y (y ≡ x + 1), and the table of P(x, y). You play ∃ (or ∀) against the computer, which plays its side perfectly. The table shows why the statement is true or false: for ∀x ∃y every row needs a T; for ∃y ∀x one column must be T all the way down. Winning strategy spells out the winning moves. Click cells to make your own predicate.