Seeing overlap with sets, flipping true and false with propositions
Split numbers that satisfy two conditions into a Venn diagram, and see exactly which region set operations like union and intersection point to. Then look at how a proposition's converse, inverse, and contrapositive relate to each other.
A set is a collection of everything that satisfies some condition. A Venn diagram draws these sets as circles, showing at a glance which region is the intersection (satisfying both conditions at once) and which is the union (satisfying either one).
A proposition is a statement that can clearly be judged true or false. Given a proposition "if p then q," swapping the condition and conclusion gives the converse, negating both gives the inverse, and doing both gives the contrapositive. Interestingly, the original proposition and its contrapositive always share the same truth value — if one is true, the other must be true too.
Change the conditions in the Venn diagram and see how the region changes, then pick a proposition example and compare how the truth or falsehood of all four statements plays out.
Set operations — in the universal set U = {1, 2, …, 12}, pick a condition for A and for B. The numbers are placed automatically in the Venn diagram, and the numbers matching whichever operation you pick below are highlighted in violet.
Result of the selected operationEverything else
Condition for A
Condition for B
Converse, inverse & contrapositive — given a proposition "if p then q" (the original), swapping the order gives the converse (if q then p), negating both gives the inverse (if not p then not q), and doing both gives the contrapositive (if not q then not p). Pick an example below and compare the truth of all four statements.