Theorem and Informal Techniques

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Theorem and Informal Techniques

• In formal logic we proved arguments that were
  universally true by their internal structure -
  arguments that were tautologies (propositional
  logic) or valid wffs (predicate logic).
• But, we also need to prove arguments that are not
  universally true, just within some context.
• Subject-specific facts become additional
  hypotheses for the proof that P ?Q in this
  particular context
• There is no formula for constructing proofs.
• Theorems are often stated and proved in a less
  formal way.

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Background

• Conjectures (hypothesis, conclusions) is a
  sentence explaining a general pattern within
  similar math problems. Like an explanation.
• Deductive (or logical) Reasoning is the process
  of demonstrating that if certain statements are
  accepted as true, then other statements can be
  shown to follow from them.
• Inductive Reasoning is the process of observing
  data, recognizing patterns, and making
  generalizations from the observations.
• Counter Example---The example disproves the
  conjectures

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Background (cont.)

• A single counter example to a conjecture is
  sufficient to disprove it.
• N2gt4N (N is an integer)
• Hunting for a counter example and being
  unsuccessful does not constitute a proof that the
  conjecture is true.
• A Famous conjecture-- Goldbach's Conjecture
• Every even n gt 2 is the sum of two primes.

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Background (cont.)

• Example of a conjecture
• 21 - 12 9
• 83 - 38 45 52 -
  25 27 64 - 46 18
• Conjuncture--The difference of a two digit
  number and its reverse is equal to a multiple of
  nine.

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Deductive Reasoning

• Moves from the general to the more specific.
• informally called a "top-down" approach.
• Try to verify the truth or falsity of a
  conjecture.
• Produce a proof P?Q
• Another option is to find a counterexample that
  disprove the conjecture.

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Inductive Reasoning

• Moves from specific observations to broader
  generalizations and theories.
• Informally, we sometimes call this a "bottom up"
  approach.
• Steps
• specific observations and measures,
• detect patterns and regularities
• formulate some tentative hypotheses that we can
  explore,
• end up developing some general conclusions or
  theories.

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Exhaustive Proof

• Use it when the conjecture is an assertion about
  a finite collection.
• The conjecture can be proved true by showing that
  is true for each member of the collection.
• Example
• For any positive integer less than or equal to
  3, the square of the integer is less than or
  equal to the sum of 10 plus 5 times the integer.
• n nn 105n
• 1 1 15
• 2 4 20
• 3 9 25

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Direct Proof

• Assume that the hypothesis P is true and deduce
  the conclusion Q.
• A formal proof require a proof sequence leading
  from P to Q.
• Example Prove that the product of two even
  integers is even.
• Let x 2m and y 2n, where m and n are
  integers. Then xy (2m)(2n)
• 2(2mn), where 2mn is an integer. Thus xy has the
  form 2k, where k is an
• integer, and xy is therefore even.

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Contraposition

• Using the tautology
• (Q?P)?(P?Q)
• We can use it to prove (P?Q)
• By proving the theorem Q ? P we can conclude
  Q?P.
• The converse (Q?P) is not equivalent.

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Contraposition (cont.)

• When theorems are stated as P if and only if Q
  we need to prove
• (P?Q) (Q?P)
• The truth of one does not imply the truth of the
  other.
• For example, if we need to prove that the product
  xy is odd if and only if both x and y are odd
  integers, we need to prove ..
• if x and y are odd, so is xy.
• if xy is odd, both x and y must be odd. ?
• (x and y are odd) ? (xy is odd)

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Contraposition (cont.)

• Proof by case
• A form of exhaustive proof.
• It involves identifying all the possible cases
  consistent with the given information and then
  proving each case separately.
• Example xy odd ? x odd and y odd
• Use contraposition (x odd and y odd) ? (xy
  odd)
• De Morgan x even or y even ? xy even.
• x even, y odd x 2m, y 2n1, and then
  xy(2m)(2n1) (2)(2mnm), which
• is even
• x odd, y even x 2m1, y 2m, and then
  xy(2m1)(2n) (2)(2mnm), which
• is even
• x even, y even x 2m, y 2n, and then
  xy(2m)(2n)(2)(2mn) where 2mn is an
• integer. Thus xy has the
  form 2k, where k is an integer, and xy is
• therefore even.

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Contradiction

• We are trying to prove P?Q. We can use the
  tautology (PQ?0)?(P?Q) to prove (P?Q). (0 stand
  for any contradiction)
• It is sufficient to prove (PQ?0)
• We need to assume that the hypothesis and the
  negation of the conclusion are true and then try
  to deduce some contradiction from this
  assumptions.
• Especially useful if you need to prove something
  is not true.

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Contradiction (cont.)

• Examples.
• Sqrt(2) is not a rational number.
• Prove by contradiction.
• Definition r is rational if r p/ q where p
  q are INTEGERS AND
• 1. p q have no common factors 2.
  q caint be 0 (of course!)
• So Lets Sqrt(2) p / q
• --- Square both sides to 2 p-sq / q -sq
• --- Quickly you see 2 divides p-sq
• --- If 2 divides p-sq, it must divide p
• --- A lil arith shows q-sq 2 x (for some x)
  which means 2 divides q
• --- Now, 2 divides BOTH p and q proveg a
  CONTRACTION re common factor

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Assignment of Olden Days

• Exercise 2.1 --- 4, 36, 47, 51
• 4 a. n is odd IFF 3n5 is odd ... Clearly
  3(2m1)56m8 IS even
• BORE --- in reverse order 3n56
  leads to n as a FRACTION, i.e., not
• an odd number (at all) FAIL IFF
  because only 1 direction holds
• b. Case is similar works in forward
  direction fails with
• 3n210 where n 8/3 which is NOT
  an integer
• 36 Prove cube-root of n is not rational Same
  idea get to
• 2 q-cube p-cube so 2 divides p
  Shortly, yu get 2 q-cube8 X
• So q-cube can be divided by 2 and
  hence q itself CONTRADICTION
• 47 For n even n gt 2, (2 to-the-n-th)-1 aint
  PRIME Get 2 to-the-2k (even
• assumption make it (2 to-the-k-th)
  squared --- Substitute see xx-1 which is
• (x-1) times (x1) so 2 to-the-n-th
  is a product of TWO distinct numbers .. (not
• itself and 1 ONLY! (This is a direct
  proof)
• 51 Prove rat irrat irrat Prove by
  CONTRADICTION Let rat irrat rat
• Almost trivial move the leftmost rat
  to the riot and see irrat MUST BE rat