Finishing 18.090 is a milestone. You will have written hundreds of proofs. You will have internalized the difference between "necessary" and "sufficient." You will wince when a friend says, "Well, it works for n=1, so it's probably true."
Prove that if $n$ is an integer and $n^2$ is even, then $n$ is even.
The hardest part of 18.090 to replicate is the blackboard defense. Find a study partner. You write a proof. They try to break it. Do not accept your own proof until your partner has failed to find a loophole. 18.090 introduction to mathematical reasoning mit
Injective (one-to-one), surjective (onto), bijective, and inverse functions. Equivalence relations (reflexive, symmetric, transitive) and partitions.
Students desiring additional experience with mathematical proofs before venturing into demanding core requirements like 18.100 (Real Analysis), 18.701 (Algebra I), or 18.901 (Topology). Finishing 18
For the student standing at the threshold of advanced mathematics, 18.090 is the key that unlocks the door. Behind that door is a universe of infinite precision, elegant abstraction, and rigorous beauty. Turn the key. The proof awaits.
The course departs from lecture-only formats. Common practices include: The hardest part of 18
Physics uses math as a tool. You are comfortable with hand-waving and infinitesimals. Mathematics demands absolute precision. 18.090 will rewire your brain.