Definition MATH Benjamin V.C. Collins, James A. Swenson MATH 2730

Transcription

1 MATH 2730 Benjamin V.C. Collins James A. Swenson

2 s and undefined terms The importance of definition s matter! may be more important in Discrete Math than in any math course that you have had previously. In order to prove something, we need to know exactly what we re talking about.

3 s and undefined terms s of square Example A polygon is called a square provided that it has four equal sides. A polygon is called a square provided that it has four equal sides and four equal angles. A polygon is called a square provided that it has exactly four equal sides and four equal angles. A polygon is called a square provided that it has four equal sides and four right angles.

4 s and undefined terms What does a definition look like? Example A polygon is called a square provided that it has exactly four equal sides and four equal angles. format [Type of thing we are talking about] is called [term being defined] provided that [accurate, complete description]. Problem But what is a polygon? What s a side? What s an angle? What s four?

5 s and undefined terms Undefined terms Remark No language can define everything. At some level, there are terms that we know, that we don t have to define. In Discrete Math, some specific concepts are undefined: natural numbers: 0, 1, 2, 3, 4,... integers:..., 3, 2, 1, 0, 1, 2, 3,... addition, subtraction, multiplication greater than / less than See the full list in the appendix!

6 s and undefined terms Let a and b be integers. We say that a is divisible by b provided that there is an integer c such that a = bc. Example 12 is divisible by 3, because 4 is an integer, and 12 = is not divisible by 5, because if c is an integer, then 12 5c.

7 s and undefined terms Language for divisibility Synonyms The following mean the same thing: a is divisible by b b divides a b is a factor of a b is a divisor of a a is a multiple of b b a Warning The expression 3 12 is a sentence, which is true. It is not proper to write 3 12 = 4.

8 s and undefined terms A quick quiz True or false? 5 30 " 7 30 % 7 30 " % 5 0 " 0 5 % 0 0 "

9 s and undefined terms Zero and divisibility Any integer b divides 0, because 0 = bc has an integer solution c. 0 does not divide any non-zero integer b, because b = 0c does not have an integer solution c.

10 s and undefined terms Even numbers An integer n is even provided that n is divisible by 2. Question Is 0 even? There is an integer k such that 0 = 2k; namely, k = 0. Thus 0 is divisible by 2. By definition, 0 is even. By the definition of even, 0 is even. Is 4x + 2y 6 even? It depends! But let s assume that x and y are integers. There is an integer k such that 4x + 2y 6 = 2k; namely, k = 2x + y 3. Thus 4x + 2y 6 is divisible by 2. So 4x + 2y 6 is even.

11 s and undefined terms Things we don t know yet Warning The following facts about even numbers are not part of the definition: Even numbers always end with 0, 2, 4, 6, or 8. Every integer that ends with 0, 2, 4, 6, or 8 is even. An even number plus an even number is an even number. An even number times any integer is an even number. etc.

12 s and undefined terms Odd numbers An integer n is odd provided that there is an integer k such that n = 2k + 1. Question Is 0 odd? There is no integer k such that 0 = 2k + 1, so 0 is not odd. Is 4x 2 + 4x + 1 odd? It depends! But let s assume that x is an integer. There is an integer k such that 4x 2 + 4x + 1 = 2k + 1; namely, k = 2x 2 + 2x. By definition of odd, 4x 2 + 4x + 1 is odd.

13 s and undefined terms More things we don t know yet Warning The following facts about odd numbers are not part of the definition: Odd numbers always end with 1, 3, 5, 7, or 9. Every integer that ends with 1, 3, 5, 7, or 9 is odd. Every integer is even or odd, but not both. An odd number plus an odd number is an even number. etc.

14 s and undefined terms Prime An integer p is called prime provided that p > 1 and the only positive factors of p are 1 and p. 8 is not prime, because: is prime, because... 2 is not prime, because 2 1. Warning 1 is not a prime number!

15 s and undefined terms Composite An integer a is called composite provided that there is an integer b such that 1 < b < a and b a. 8 is composite, because: < 2 < 8 7 is not composite, because: 2 7, 3 7, 4 7, 5 7, and is not prime, because 2 1. Remark 1 is neither prime nor composite.