Honors Discrete Mathematics
Honors Discrete Mathematics MAD 2104
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This 7 page Class Notes was uploaded by Dino Corwin on Monday October 12, 2015. The Class Notes belongs to MAD 2104 at Florida Atlantic University taught by Jorge Viola-Prioli in Fall. Since its upload, it has received 12 views. For similar materials see /class/221637/mad-2104-florida-atlantic-university in Mathematics Discrete at Florida Atlantic University.
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Date Created: 10/12/15
2DEFINITION Discrete Mathematics is the Math needed in decisionmaking in non continuous situations Thus it mainly deals with discrete objects their best examples being the nite sets However we will have to consider in nite sets as well In broad terms think of discrete objects as objects that can be separated from one another The material we will discuss differs fundamentally from the topics of a typical Calculus course as the realm of Calculus is the continuous J Viola Prioli It can be said that such a course can be thought of as a dramatic production But then who are the characters The three main characters are De nition Theorem and Proof there is also an quotevilquot character to be mentioned at a later stage A mathematical de nition must be absolutely precise whether it is an object a concept a command etc with no room for ambiguity De nitions are not to be argued about they are accepted as long as they do not contradict what is known to be true In Mathematics is crucial to understand a de nition before proceeding In what follows we assume knowledge of the set of integers denoted by Z and the three basic operations x and the order relations lt s gt 2 The natural numbers denoted by N are the nonnegative integers therefore N 0 1 2 3 whereas Z 2 1 O 1 2 3 J Viola Prioli NOTE some authors de ne N as the set of positive integers thus deleting 0 from N However we will adhere to our textbook s de nition Some notations and remarks are worth considering The symbol E means quotbelongs to Thus we can write 4 E N Their negation is denoted by i We use curly brackets to indicate a set of elements For simplicity we sometimes write pq or pq to denote the product pxq Fractions like 37 play almost no role in this course as we deal with integer numbers almost exclusively Next we will explore three important concepts J Viola Prioli Given integers a and b we say that a is divisible by b if there exists an integer c so that a bx c There are equivalent ways to express that a is divisible by b a is a multiple of b b divides a b is a factor of a b is a divisor of a In any case a vertical bar is the symbol to be used hence ba is read quotb divides a or any ofthe equivalent forms CAUTION 3 6 is true because 6 2x3 is a multiple of 3 Observe that 3 6 is a sentence is not a number On the other hand 36 is a fraction whose value is 5 and so is 63 which equals 2 Therefore quot648quot is true quot410quot is false quot1 a for every integer aquot is true 312 is true since 12 3gtlt 4 so here 4 c of our de nition above Also observe that a a for every integer a including a O J Viola Prioli An integer a is even if it is a multiple of 2 Therefore a2c for some integer c The even numbers are thus 4 2 O 2 4 6 8 Observe that O is even as O 2 x O An integer a is odd if a 2b 1 for some integer b The odd numbers are therefore 3 1 1 3 5 7 For instance 7 is odd because we can write 7 2 4 1 Notice that a number is odd if it comes immediately after an even number A positive integer p is called a prime if pgt1 and its only positive divisors are 1 and p itself So for p to be a prime number it must satisfy these three conditions simultaneously 1 p must be an integer 2 p must be greater than 1 3 the only positive divisors of p must be 1 and p J Viola Prioli Examples of numbers that are not primes E 1 fails 7 2 fails 187 1 fails 14 3 fails as 14 2 x 7 The rst prime numbers are 2 3 5 7 11 13 17 19 23 29 We observe that the prime numbers other than 2 are odd However 15 is odd but is not a prime number In a later chapter we will prove that there are in nitely many prime numbers A positive integer a is called composite if there exists an integer b such that 1 lt b lt a and ba In other words a admits a proper divisor Recall that every number is divisible by itself and by 1 what makes a number composite is the fact that admits other divisors Notice that composite numbers quotbreak down as a product of strictly smaller natural numbersquot if a is composite according to the de nition we have a proper factor b Hence a b c and since 1 lt b lt a it is clear that 1 lt c lt a Thus a is the product of two smaller factors J Viola Prioli Observe that according to the de nitions of this section 0 and 1 are special in that they are neither prime nor composite a Find three different prime numbers p q and 2 such that pqz 1 is prime b Find three different prime numbers a b and c such that abc 1 is composite J Viola Prioli
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