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Intro to Proofs

Free Intro to Proofs study guide: 9 units with summaries, key ideas, flashcards and 270 practice questions with step-by-step solutions.

Lines up with: College transition-to-proofs course.

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Units

  1. UNIT 01

    Logic and Truth Tables

    Propositional logic is the grammar of proof: it tells you exactly what a statement claims, when it is false, and which rewrites keep its meaning. Every later unit depends on reading conditionals and negations correctly.
  2. UNIT 02

    Quantifiers

    Quantifiers say how many objects a statement is about: all of them or at least one. Reading, writing and negating quantified statements is what lets you know exactly what a proof must show.
  3. UNIT 03

    Direct Proof

    A direct proof starts from the hypothesis, unpacks the definitions, and uses algebra and known facts to arrive at the conclusion. This is the workhorse method for statements about integers, divisibility and rational numbers.
  4. UNIT 04

    Contrapositive and Contradiction

    Indirect proofs handle statements that resist a direct attack. Contrapositive proof flips a conditional into an easier one, and contradiction shows the opposite of a claim is impossible.
  5. UNIT 05

    Sets and Set Proofs

    Sets are the basic objects of modern mathematics. This unit covers the set operations and the element-chasing technique used to prove that one set is contained in, or equal to, another.
  6. UNIT 06

    Mathematical Induction

    Induction proves a statement for infinitely many integers with two finite checks: a base case and an inductive step. It is the standard method for sums, divisibility patterns, inequalities and recursively defined sequences.
  7. UNIT 07

    Relations and Equivalence Relations

    Relations describe how elements of a set are connected. Equivalence relations sort a set into classes of 'alike' elements, and congruence modulo n is the key example.
  8. UNIT 08

    Functions: Injective, Surjective, Bijective

    Functions are the links between sets. This unit defines injective, surjective and bijective functions, shows how to prove each property, and connects them to composition, inverses and counting.
  9. UNIT 09

    Cardinality and Countability

    Cardinality measures the size of sets, including infinite ones, by matching elements with bijections. Some infinite sets, like ℕ, ℤ and ℚ, are countable, while others, like ℝ and P(ℕ), are strictly larger.