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Mathematical Induction
Assume that we are given an infinite supply of stamps of two different denominations, 3 cents and and 5 cents. Prove using mathematical induction that it is possible to make up stamps of any value 8 cents.
Basis: We can make an 8 cents stamp by using one 3-cents and one 5-cents stamp. Inductive hypothesis: Assume that we can make stamps of values k = 8, 9, 10, … n. Inductive step: Show that we can compose a stamp of value n + 1.
Composing Stamps: Example 3
Illustration
8 = 3 + 5
9 = 3 + 3 + 3 = 3 * 3
10 = 5 + 5 = 2 * 5
11 = 1 * 5 + 2 * 3
Outline of Proof
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Composing Stamps – Actual Proof
Inductive step: Show that we can compose a stamp of value n + 1.In composing n, several (or none) 3-cents and several (or none) 5-cents stamps have been used.To go from n to n + 1, we consider two cases.Case 1 : If there is at least one 5-cents stamp in the collection, replace it by two 3-cents stamps. This gives us stamps of value n + 1 cents.
Case 2 : Suppose that the current collection uses only 3-cents stamps. Since n 8, there must three 3-cents stamps in the collection. Replace these three stamps by two 5-cents stamps. This gives us a stamp of value n + 1 cents.The proof is complete. Another Proof
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Another Proof
Another Proof
n + 1 = 3 * (p + 2 ) + 5 * (q – 1) if q > 1
n + 1 = 3 * ( p – 3 ) + 5 * (q + 2 ) if p > 3
For all n > 8, n can be expressed as a linear combination of 3 and 5, that is n = 3 * p + 5 * q for n >= 8
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Review Propositional Logic – Page 1
Implication In propositional logic "if p then q" is written as p q and read as "p implies q". p q p q same as p q
F F T F T T T F F T T T
How to remember this definition?Implication is false only when the premise is true and the consequence is false.
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Propositional Logic – Page 2
Bi-conditional p q means p iff q
p q p q same as
F F T
F T F
T F F
T T T
How to remember this definition?Bi-conditional is a like a magnitude comparator, it is true when both inputs are identical.
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More Terminology
ContradictionA logical expression that is always false, regardless of what truth values are assigned to its statement variables, is called a contradiction. The statement p p is a contradiction.TheoremIf A and B are logical statements and if the statements A and A B are true, then the statement B is true.
A logical expression that is always true, regardless of what truth values are assigned to its statement variables, is called a tautology. The statement p p is a tautology.
Tautology
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Simple Logic Proofs
A B A B A (A B) A (A B) B F F T F T F T T F T T F F F T T T T T T
Prove that A (A B) B is a contradiction. A (A B) B = A (A B) B= (A A) (A B) B using distributive law.= F (A B) B = (A B) B = A (B B) using associative law = A F = F Contradiction. Hence B must be true.
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An Important Theorem
Among the four statementsp q statementq p conversep q inverseq p contra-positive
1. The statement and its contra-positive are equivalent.
2. The converse and inverse are equivalent.3. No other pairs in the statements given
above are equivalent.Proof: Make a truth table to see that 1 and 2 are tautologies.
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Proof By Contradiction
Prove that is not a rational number. (Example 1.7, page 13)2
Proof: Assume is a rational number. Let m and n be two integers, with no common factor, such that
2
m
n2
Or 222 nm
2n
22 42 km 22 2km
This shows that is even.2m
Since is a multiple of 2, it is an even number. Therefore, n is even must be of the form 2k for some integer k.
1
1
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Proof Continued …
We have concluded that both and are even, therefore, n and m are also even They must have a common factor. This contradicts our assumption. Hence is not a rational number.
2n 2m
2
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Proof of Proposition 1
p q p q ~p
~q
~q ~p (p q)(~q ~p)
F F T T T T T
F T T T F T T
T F F F T F T
T T T T T T T