Therefore the Fibonacci numbers in this sum are all different from F l . As a result, one can write m as the sum of one F l , and a few distinct Fibonacci numbers which are less than F l . We finished the induction.
Lecture 01 § 3 Definition • Definition (in mathematics) gives a precise meaning of a term. • Some typical definition forms: An object X is called the term being defined provided it satisfies specific conditions . We say X is the term being defined provided it satisfies specific conditions . • Examples: (1) Define the following terms: divisible, odd, even, prime, composite. You could use the basic concept of integers (including additions and multiplications). (2) Suppose the distance between two points in the plane is defined already. Give a defi- nition of collinear of three points. § 4 Theorem • A theorem is a declarative statement (about mathematics) for which there is a proof. • Differences between theorems, conjectures, and mistakes. • The table of “If A then B ” statement A B True True Possible True False Impossible False True Possible False False Possible • Equivalent expressions to “If A then B ”: (1) A implies B . (2) Whenever A , we have B . (3) A is sufficient for B . (4) B is necessary for A . (5) A = ⇒ B . • (Vacuous truth) Sometimes the condition A is always impossible. In this case the statement “If A then B ” is always true. • The table of “ A if and only if B ” statement A B True True Possible True False Impossible False True Impossible False False Possible • Equivalent expressions to “ A if and only if B ”: (1) A iff B . (2) A is sufficient and necessary for B . 1
(3) A is true exactly when B is true. (4) A ⇐⇒ B . • And, Or, and Not statements. § 5 Proof • Direct proof of “if A then B ” (1) Invent suitable notations and restate A . (2) Write down B . (3) Figure out a bridge between A and B . • Direct proof of “ A if and only if B ”: prove “if A then B ” and “if B then A ”. • Examples: (3) Let x be an integer. If x > 1, then x 3 + 1 is composite. (4) Suppose x, y, z are positive real numbers. Prove that x 3 + y 3 + z 3 ≥ 3 xyz . HW1(a) (Due 2/1/2016) • 3.4 • 4.2(a),(g),(h) • 5.21 2
Lecture 02 § 6 Counterexample • To disprove “if A then B ”, we just need to find an example where A is true but B is false. • Examples: (1) Disprove: If a , b , c are positive integers such that a | bc , then a | b or a | c . (2) Disprove: if p is prime, then 2 p - 1 is also prime. § 7 Boolean algebra • Boolean algebra includes expressions containing letters and operations, where each letter stands for the value TRUE or FALSE. The basic operations are ∧ and : x ∧ y is true if and only if both x and y are true. ∨ or : x ∨ y is true if and only if at least one of x and y is true. ¬ not : ¬ x is true if and only if x is false.
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- Fall '16
- Sam Liu
- Natural number, Equivalence relation