Contradiction: means to start assuming p and ¬q, and follow a path toward which a statement which cannot be true is stated as true. The beauty
of this proof is that we assert the opposite of the conclusion and as a byproduct some other statement becomes self contradictory i.e
(¬q ^ p) → (r ^ ¬r), where (r ^ ¬r) isn’t even the conclusion to the original hypothesis it’s something else that’s a result of the proof
Step 1) Assume the hypothesis and the negation of the conclusion to be true.
Step 2) Perform operations such that a conclusion is reached which is not true
For instance, a number previously stated to be odd is even (or vv), irrational to be rational (or vv), integer to be non integer, to be in
one domain and then not in that domain, to be positive and then negative.
Step 3) State that the negated conclusion causes a contradiction, thus the original conjecture is true.
Contraposition: means to negate the conclusion and show that it implies that the premise is not true. Taking advantage of the fact that p→q is
logically equivalent to ¬q → ¬p.
Step 1) Assume the negation of the conclusion
Step 2) Perform operations that lead to the fact that the hypothesis is not true.
Step 3) State that we have proven the conjecture by contraposition.
Direct proof: means that through assuming the hypothesis is true and considering all the members of its domain we may conclude that the
conclusion is true.
Step 1) Assume that the hypothesis is true.
Step 2) Perform steps which lead to asserting that the conclusion is true.
Step 3) State that we have proven the conjecture.
Proof by cases: If a domain under consideration may be divided then proofs generated for each division then the conjecture can be proven.
Proof by exhaustion: Consider every single case within a domain and prove it either true. (If one case fails, the result is false, and that case will
give you the counterexample)
Vacuous Proof: Prove that a conjecture p→q is always true because p is always false.
Trivial Proof: Proving that a conjecture p→q is always true because its q is true regardless of p.
(Dis)Proof by Counterexample: Finding a value that contradicts the conjecture (by showing that P is true and Q is false for that value).
Constructive proof: Finding a value for an existence proof.
Non-constructive proof: Showing that a number has the property of the thing trying to be proved and then once that value is asserted, using it to
prove the second portion of the conjecture.
Proofs of equivalence: When conjectures are compound propositions, we may prove or disprove them by proving or disproving their
equivalencies.
If and only if (↔) : In order to prove this you must prove p→q and q→p (p←q) are both true using the above methods.
Uniqueness: Is generally performed by contradiction. I may involve solving for the unique value and then trying to produce similar values by
the same method that leads to a contraction in the character of those values (some violation of the type of number they are).
Irrational Number proofs: In general proving results involving irrational numbers requires a proof by contradiction; sometimes it requires a
proof by contraposition.
Rules for integers:
If an integer is added, multiplied or subtracted by another the result is an integer.
Oddness: An integer n is odd if n = 2*k +1, where k is an integer
Evenness: An integer n is even if n = 2*k, where k is an integer
Rules for rational numbers:
A number is rational if it can be written in the form a/b where a and b are integers and b is non-zero (if a and b have no common factors then it
is in lowest terms, but a rational does not have to be in lowest terms).
Rational numbers may be added, multiplied, divided and subtracted to other rational and integer and produce rational numbers. Integers are
contained within the rational numbers, and natural numbers are contained within the integers.
Irrational numbers are not rational. √2 is an example of an irrational number. Irrational numbers may be added to, subtracted, multiplied and
divided by rational numbers and produce irrational numbers. However their interaction with irrational numbers produces a number of different
effects.
Absolute value definition: |a|, the absolute value of a, equals a when a ≥ 0 and equals –a when a ≤ 0 (which is always the positive value)
Perfect square: An integer a is a perfect square if there is an integer b such that a = b 2