3.DEFINITION. examples being the finite sets. However, we will have to consider infinite sets as well.

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1 3.DEFINITION Discrete Mathema6cs is the Math needed in decision making in noncon6nuous situa6ons. Thus, it mainly deals with discrete objects, their best examples being the finite sets. However, we will have to consider infinite 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 conbnuous.

2 It can be said that such a course can be thought of as a dramabc producbon. But then, who are the characters? The three main characters are: DefiniBon, Theorem and Proof. (there is also an "evil" character, to be menboned at a later stage) A mathema6cal defini6on must be absolutely precise (whether it is an object, a concept, a command, etc) with no room for ambiguity. DefiniBons are not to be argued about: they are accepted as long as they do not contradict what is known to be true. In Mathema6cs is crucial to understand a defini6on before proceeding. In what follows we assume knowledge of the set of integers (denoted by Z) and the three basic operabons: +,, and the order relabons <,, >, The natural numbers (denoted by N) are the non negabve integers: therefore, N= {0, 1, 2, 3,...} whereas Z= {...., 2, 1, 0, 1, 2, 3,...}

3 NOTE: some authors define N as the set of posibve integers, thus delebng 0 from N. However, we will adhere to our textbook's definibon. Some notabons and remarks are worth considering: The symbol means "belongs to". Thus, we can write 4 N. Their negabon is denoted by. We use curly brackets { } to indicate a set of elements. For simplicity we somebmes write p.q (or pq) to denote the product p q FracBons (like 3/7) play almost no role in this course, as we deal with integer numbers almost exclusively. Next we will explore three important concepts.

4 DEFINITION. Given integers a and b we say that a is divisible by b if there exists an integer c so that a = b c. There are equivalent ways to express that a is divisible by b: a is a mulbple of b b divides a b is a factor of a b is a divisor of a In any case a verbcal bar is the symbol to be used: hence, b a is read "b divides a" or any of the equivalent forms. CAUTION: 3 6 is true (because 6 = 2 3 is a mulbple of 3). Observe that 3 6 is a sentence, is not a number! On the other hand, 3/6 is a fracbon (whose value is.5) and so is 6/3, which equals 2. Therefore "6 48" is true, "4 10" is false, "1 a for every integer a" is true, 3 12" is true (since 12 = ( 3) ( 4), so here 4 = c of our definibon above). Also observe that a a for every integer a, including a = 0.

5 DEFINITION. An integer a is even if it is a mulbple of 2. Therefore, a=2c for some integer c. The even numbers are thus {..., 4, 2, 0, 2, 4, 6, 8,...}. Observe that 0 is even, as 0 = 2 0. DEFINITION. 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. NoBce that a number is odd if it comes immediately ajer an even number. DEFINITION. A posibve integer p is called a prime if p>1 and its only posibve divisors are 1 and p itself. So for p to be a prime number it must sabsfy these three condibons simultaneously: 1) p must be an integer 2) p must be greater than 1 3) the only posibve divisors of p must be 1 and p.

6 Examples of numbers that are not primes: (1 fails), 7 (2 fails), 5 18/7 (1 fails), 14 (3 fails as 14 = 2 7) The first 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 infinitely many prime numbers. DEFINITION. A posibve integer a is called composite if there exists an integer b such that 1 < b < a and b a. 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. NoBce that composite numbers "break down as a product of strictly smaller natural numbers": if a is composite, according to the definibon we have a proper factor b. Hence a = b c and since 1 < b < a it is clear that 1 < c < a. Thus, a is the product of two smaller factors.

7 Observe that according to the definibons of this secbon, 0 and 1 are special in that they are neither prime nor composite. EXERCISE a) Find three different prime numbers p, q and z such that pqz + 1 is prime. b) Find three different prime numbers a, b and c such that abc + 1 is composite