Read and Write Mathematical Proofs

Mathematical prose is a compressed dialect; learn to decompress logic and set notation before the proofs start speaking it.

And, or, not, implies: four words carrying all of mathematics, and the truth table is where their meaning is settled for good.

Sets are the nouns of mathematics and the conditional is its most misread verb; both deserve one careful pass.

Almost every interesting statement in mathematics alternates quantifiers; misplace one and you have proved something else.

Bounded quantifiers and set operations are where the logic meets the mathematics you already know.

Every proof you will ever write opens with a move from this table; knowing which move the goal demands is half the craft.

Let x be arbitrary versus choose x: the two openings that handle every for-all and there-exists you will ever face.

Proof by cases and existence-with-uniqueness are the last two strategy patterns; after these the table is complete.

Worked proofs with the strategy labels removed, the way you will meet them in every book from now on.

Ordered pairs, then relations and their properties: the machinery under every "is related to" in mathematics.

Partial orders rank things and equivalence relations sort them into bins; the partition theorem is the first structure theorem you prove.

Injective and surjective are the two adjectives the rest of mathematics never stops using; prove them, do not vibe them.

When a function can be undone, and what it does to whole sets at once: the working grammar of higher mathematics.

The domino argument, stated once as a principle and once as a proof technique you can execute under exam pressure.

Recursive definitions and inductive proofs are the same idea seen from two sides; sums and sequences make it concrete.

Sometimes the previous domino is not enough and you need all of them; strong induction is the honest way to say so.

Euclid's algorithm and unique factorization are the first two theorems most mathematicians ever loved; prove them properly.

Clock arithmetic made rigorous, plus Euler's theorem: the pure mathematics that will pay off one competency from now.

RSA is number theory cashing every check the last two competencies wrote; a proof-based idea guarding actual money.

Cantor's diagonal is the most famous proof in mathematics and the final exam of your bijection skills.

Two injections buy you a bijection, and the axiom of choice is where the ground under mathematics is quietly load-bearing.