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Logic and Proof Method Diagnostics

Retrieval Prompts​

  1. State the negation rules for universal and existential quantifiers from memory.
  2. Explain the difference between implication, converse, and contrapositive.
  3. List the situations in which you would first consider direct proof, contrapositive, contradiction, cases, and counterexample.
  4. Define logical equivalence without using the word "similar."

Compare and Distinguish​

Separate these pairs clearly:

  • for all x exists y P(x, y) versus exists y for all x P(x, y)
  • contrapositive proof versus contradiction proof
  • proving P -> Q versus proving P <-> Q
  • invalid example-checking versus valid counterexample construction

Common Mistake Check​

For each statement below, identify the error:

  1. "The negation of for all x P(x) is for all x not P(x)."
  2. "I proved Q -> P, so P -> Q is done."
  3. "I checked the statement for n = 1, 2, 3, 4, so it must be true for all n."
  4. "I used contradiction because I was not sure what else to do."

Mini Application​

For each statement, do all three tasks:

  1. classify the outer logical form
  2. negate it correctly
  3. choose the first proof or disproof method you would try

Statements:

  1. Every even integer greater than 2 is the sum of two primes.
  2. If A subseteq B, then A intersect C subseteq B intersect C.
  3. There exists an integer n such that n^2 = 2.
  4. A real number is nonzero if and only if it has a multiplicative inverse.
  5. Every relation on a finite set is transitive.

Evidence Check​

This page is complete only if you can explain your method choices in sentences, not just write labels like "contradiction" or "direct."

Integrated Logic Drill​

Translate each English statement into symbolic form, define the domain, then test the claim with either a truth table, a counterexample, or a short proof sketch.

  1. If a program accepts empty input, then it must either return a default value or raise a documented error.
  2. Every user has at least one role, but not every role belongs to an active user.
  3. A test suite is insufficient when there exists a public behavior with no assertion covering it.
  4. If no request mutates shared state, then all requests commute with each other.

For each statement:

  • name the propositions or predicates
  • state whether implication, conjunction, disjunction, negation, or quantification is doing the main work
  • write the negation in plain English
  • give one concrete counterexample if the statement is false