A set is "a collection of things with a clear boundary." Anything where you can clearly tell whether an object belongs or not — "multiples of 2," "even numbers," "the students in our class" — can form a set. Given two sets A and B, there are a few different ways to combine them, and drawing a Venn diagram (two overlapping circles) makes all of them much more intuitive.
The union A∪B collects everything that belongs to A or to B (counting any overlap only once). The intersection A∩B is only what belongs to both A and B at the same time — the overlapping part. The difference A−B is what's in A but not in B, and the complement Aᶜ is everything in the universal set that isn't in A.
There's a formula that matters a lot when counting sets: |A∪B| = |A| + |B| − |A∩B|. If you just add the number of elements in A and the number in B, the overlapping part (the intersection) gets counted twice. So you have to subtract it once to get the correct size of the union. This idea turns out to be the foundation of the "addition rule" you'll use later when counting outcomes.
A proposition is a statement whose truth or falsehood can be clearly determined. "3 is a prime number" is a true proposition, while "this sentence is interesting" isn't a proposition at all, since different people would judge it differently. Given a proposition of the form "if p then q" (the original), three other propositions can be derived from it: the converse (if q then p), which swaps the order of p and q; the inverse (if not p then not q), which negates both; and the contrapositive (if not q then not p), which does both — swaps the order and negates.
Here's a really important fact: the original proposition and its contrapositive always share the same truth value. If "if it's even, then it's an integer" is true, its contrapositive, "if it's not an integer, then it's not even," must also be true. The converse and the inverse, on the other hand, can have a different truth value from the original (though the converse and inverse are each other's contrapositive, so they always share a truth value with each other). Thanks to this property, when the original proposition is hard to prove directly, it's just as good to prove that its contrapositive is true instead. This is called proof by contrapositive.
When studying this with kids, it helps to start with the question "is the converse always true?" "If it rains, the ground gets wet" is true, but its converse, "if the ground is wet, it rained," might not be (someone could have sprayed it with a hose). Starting with an everyday example like this, to let the difference between a proposition and its converse sink in, makes the jump to mathematical examples feel much more natural. On our activity page, you can pick number conditions yourself and check set operations on a Venn diagram, then compare the truth of the original, converse, inverse, and contrapositive at a glance across several proposition examples.