Basic properties of the Integers

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1 Basic properties of the Integers Branko Ćurgus May 2, Axioms for the Integers In the axioms below we use the standard logical operators: conjunction, disjunction, exclusive disjunction, implication, universal quantifier, existential quantifier. We also use the standard set notation: set membership, subset, equality =, negation of equality, the empty set, Cartesian product. For singleton sets instead of writing {a} = {b} we write a = b. The notation f : A B stands for a function f which is defined on a set A with the values in B. Axiom 1 below establishes the existence of the addition function defined on Z Z with the values in Z. It is common to denote the value of + at a pair a,b Z Z by a+b. Axiom 6 establishes the existence of the function multiplication function defined on Z Z with the values in Z. It is common to denote the valueof this function at a pair a,b Z Z by a b which is almost always abbreviated as ab. Axiom 12 introduces a strict total order < on Z. In Axiom 16 we use the abbreviation a b for the proposition a < b a = b. This axiom is the subtlest of all the axioms. Axiom 0 ZE. Z is a nonempty set. Axiom 1 AE. There exists a function + : Z Z Z. Axiom 2 AA. a Z b Z c Z a+b+c = a+b+c Axiom 3 AC. a Z b Z a+b = b+a Axiom 4 AZ. 0 Z a Z 0+a = a 1

2 Axiom 5 AO. a Z x Z a+x = 0 Axiom 6 ME. There exists a function : Z Z Z. Axiom 7 MA. a Z b Z c Z abc = abc Axiom 8 MC. a Z b Z ab = ba Axiom 9 MO. 1 Z 1 0 a Z 1 a = a Axiom 10 MZ. a Z b Z ab = 0 a = 0 b = 0 Axiom 11 DL. a Z b Z c Z ab+c = ab+ac Axiom 12 OE. a Z b Z a < b a = b b < a Axiom 13 OT. a Z b Z c Z a < b b < c a < c Axiom 14 OA. a Z b Z c Z a < b a+c < b+c Axiom 15 OM. a Z b Z c Z a < b 0 < c ac < bc Axiom 16 WO. S Z S x S 0 < x m S x S m x Explanation of abbreviations: ZE - integers exists, AE - addition exists, AA - addition is associative, AC - addition is commutative, AZ - addition has zero, AO - addition has opposites, ME - multiplication exists, MA - multiplication is associative, MC - multiplication is commutative, MO - multiplication has one, MZ - multiplication respects zero, DL - distributive law, OE - order exists, OT - order is transitive, OA - order respects addition, OM - order respects multiplication, WO - the well-ordering axiom. The following implications will be used in the proofs without being mentioned explicitly: a Z b Z c Z a = b b = c a = c 1.1 a Z b Z c Z a = b b < c a < c 1.2 a Z b Z c Z d Z a = b c = d a+c = b+d 1.3 a Z b Z c Z d Z a = b c = d ac = bd 1.4 Implication 1.1 follows from the implication A = B B = C A = C which is easily proved for sets A,B,C. Implication 1.2 is proved by contradiction. First we form the negation of 1.2: a Z b Z c Z a = b b < c c = a c < a. 1.5 Next we prove that 1.5 implies a false proposition. Since the conjunction is distributive with respect to the exclusive disjunction, 1.5 is equivalent to the quantifiers are dropped to save the space a = b b < c c = a a = b b < c c < a

3 Using the commutativity of conjunction and 1.1, the proposition 1.6 implies b = c b < c a = b b < c c < a. By Axiom OT the last proposition implies b = c b < c a = b b < a. 1.7 Further, Axiom OE implies that both b = c b < c and a = b b < a are false. Thus 1.7 is false. Since we proved that 1.5 implies 1.7 is true, we must have that 1.5 is false. This proves that 1.2, the negation of 1.5, is true. The implications 1.3 and 1.4 follow from the fact that the addition and the multiplication are functions. 2 Basic algebraic properties of the integers In this section we list properties of the integers which involve the axioms related to the addition and the multiplication, but not the order. Proposition 2.1. The element 0 Z introduced in AxiomAZ is unique. Proof. Assume that there exist 0 Z such that 0 + a = a for all a Z. Since 0 Z universal instantiation yields, = 0. Applying universal instantiation with 0 Z to AxiomAZ we get = 0. Since by AxiomAC and universal instantiation, we have 0 +0 = 0+0. The last three equalities yield 0 = 0. Proposition 2.2. For every a Z the equation a+x = 0 has a unique solution. Proof. Let a Z be arbitrary. By Axiom AO the equation a + x = 0 has a solution. Assume that a + x = 0 and a + x = 0. Since addition is a function, a + x = 0 implies x +a+x = x +0. Using AxiomsAA andaz we deduce x +a+x = x. By assumption a+x = 0, hence 0+x = x, and by AxiomAZ x = x. Definition 2.3. Let a Z. The unique solution of the equation a+x = 0 is denoted by a and called the opposite of a. For b Z we write b a instead of b+ a. Proposition = 0. Proof. By Definition is the unique solution of the equation 0 + x = 0. Since by AxiomAZ we have 0+0 = 0, we deduce 0 = 0. Proposition 2.5. For every a Z we have a0 = 0a = 0. 3

4 Proof. Let a Z be arbitrary. By AxiomAZ and universal instantiation we have 0+0 = 0. Since the multiplication is a function a0 + 0 = a0. By Axiom DL a0 + a0 = a0. By Axiom ME a0 Z. Hence a0 Z exists by Axiom AO. Now a0 + a0 = a0 yields a0+a0 a0 = a0 a0. By AxiomAA andao we obtain a0 = 0. AxiomAC now yields a0 = 0a = 0. Proposition 2.6. Let a,b and c be integers. Then a+c = b+c implies a = b. Proof. Exercise. Proposition 2.7. For every a Z we have a = a. Proof. Let a Z be arbitrary. By definition a is the unique solution of the equation a + x = 0. Since by definition of a we have a + a = 0 and since by Axiom AC a+a = a+ a = 0, we see that a is a solution of a+x = 0. By Proposition2.2 we conclude that a = a. Proposition 2.8. For every a Z and for every b Z we have ab = b a = ab. Proof. Let a and b be be arbitrary integers. Then by AxiomAO we have a+ a = 0. By AxiomsME, we have a+ a b = 0b. Now, AxiomsMC anddl and Proposition2.5 yield ab + ab = 0. Thus, ab is a solution of ab + x = 0. By Definition2.3, this implies ab = ab and AxiomMC yields ab = b a = ab. Proposition 2.9. For every a Z and for every b Z we have a b = ab. Proof. Let a and b be arbitrary integers. By Proposition2.8 we have a b = a b. Applying Proposition2.8 yields a b = ab. Hence a b = ab. Now, Proposition2.7 implies ab = ab, and consequently a b = ab. 3 Basic properties of the integers involving the order Proposition 3.1. Let a and b be integers. Then a < b if and only if b < a. Proof. Let a and b be arbitrary integers. Assume a < b. Since a+ b is an integer, by AxiomOA we deduce a+ a+ b < b+ a+ b. By AxiomsAA and AC we conclude a+ a + b < b+ b + a. Now, by the definition of an opposite and AxiomsAZ we conclude b < a. The converse b < a a < b is proved similarly. Proposition 3.2. Let a,b and c be integers. Then a < b and c < 0 imply bc < ac. 4

5 Proof. Let a,b and c be arbitrary integers. Assume a < b and c < 0. Then by Propositions 3.1 and < c. Now, a < b, 0 < c and AxiomOM imply a c < b c. By Proposition 2.8, the lase inequality can be rewritten as ac < bc. By Propositions 3.1 the last inequality implies bc < ac. Proposition 3.3. Let a,b and c be integers. Then ac < bc and 0 < c imply a < b. Proof. Leta,bandcbearbitraryintegers. Noticethatthispropositionhastheformp q r which is equivalent to r q p. Therefore we will prove a b 0 < c ac bc. since a b means a > b or a = b we will proceed with two cases. If a > b, then, since 0 < c, by AxiomOM we conclude ac > bc and thus also ac bc. In the second case, if a = b, then AxiomME, that is the fact that the multiplication is a function, implies that ac = bc. So, ac bc in this case as well. Proposition 3.4. Let a and b be integers. Assume 0 < a. Then 0 < b if and only if 0 < ab. Proof. Let a and b be arbitrary integers. Throughout this proof we assume that 0 < a. We first prove 0 < b 0 < ab. Assume 0 < b. Since 0 < a and 0 < b, AxiomOM implies 0b < ab. Since by Proposition 2.5 0b = 0, we have 0 < ab. To prove the converse assume that 0 < ab. Proposition 2.5 and AxiomMC yield 0a < ba. Since 0 < a, Proposition 3.3 implies 0 < b. Definition 3.5. For a Z the product aa is called the square of a and it is denoted by a 2. Proposition 3.6. For every nonzero integer a we have 0 < a 2. Proof. Let a be an arbitrary nonzero integer. By Axiom OE we have two possibilities for a: either 0 < a or a < 0. Case 1. Assume 0 < a. By Propositions 3.1 and 2.4 we have that a < 0. Since 0 < a, Axiom OM yields a a < 0 a. By Propositions 2.5 and 2.8 we deduce a a < 0. By Propositions 3.1 and 2.4 we deduce 0 < a 2. Case 2. Assume a < 0. By Propositions 3.1 and 2.4 we have that 0 < a. Since 0 < a, Axiom OM yields 0 a a < 0 a. By Propositions 2.5 and 2.8 we deduce a a < 0. By Propositions 3.1 and 2.4 we deduce 0 < a 2. Corollary < 1. Proof. By Axiom MO we have 1 0. By Proposition 3.6 we deduce 0 < 1 2. By Axiom OM 1 2 = 1. Thus, 0 < 1. A different proof proceeds by contradicting. Assume 1 < 0. Then, by Propositions 2.4 and 2.7, we have 0 < 1. Since 1 < 0 and 0 < 1, Axiom OM implies 1 1 < 0 1. By Axiom AO and Proposition 2.5 we deduce 1 < 0. By Propositions 2.4 and 2.7 the last inequality yields 0 < 1. Thus, we proved that 1 < 0 implies 0 < 1 is true. The implication p p can be true only if p is false. Hence 1 < 0 is false. That is 0 < 1 is true. 5

6 Corollary 3.8. For every a Z we have a < a+1. Proof. Let a Z be arbitrary. Since 0 < 1, Axiom OA implies 0+a < 1+a. By Axioms AZ and AC we have a < a+1. By Corollary 3.8, 1 < 1+1. Therefore we define Definition = 1+1. By Corollary 3.8, 2 < 2+1. Therefore we define Definition = 2+1, 4 = 3+1, 5 = 4+1, 6 = 5+1, 7 = 6+1, 8 = 7+1, 9 = 8+1. By Corollary 3.8, 0 < 1 < 2 < 3 < 4 < 5 < 6 < 7 < 8 < 9. Exercise Prove 2+2 = 4. Exercise Prove 2 2 = 4. Definition Let a Z. If a > 0, then a is said to be positive. If a < 0, then a is said to be negative. We define Z + = { y Z 0 < y } and Z = { x Z x < 0 }. 4 The Well Ordering Axiom Definition 4.1. Let S be a nonempty subset of Z. We say that S has a minimum if there exists m S such that for every x S we have m x. Formally, S has a minimum m S x S m x. 4.1 The integer m S satisfying the proposition on the right-hand side of 4.1 is called the minimum of S. It is denoted by mins. With this definition the well ordering AxiomWO can be restated as S Z + S S has a minimum 4.2 Recall that the propositions p q r and p r q are equivalent. Consequently, the well ordering AxiomWO, as stated in 4.2, is equivalent to S Z + S does not have a minimum S = 4.3 At this point it is useful to note the formal meaning of the phrase S does not have a minimum. Negating 4.1 we get: S does not have a minimum x S y S y < x. I will illustrate how to use 4.3 in the following proposition. 6

7 Proposition 4.2. There are no integers between 0 and 1. Proof. Define a set S by S = {x Z 0 < x x < 1}. Clearly S Z +. Next we will prove that S does not have a minimum. Let x S be arbitrary. Then 0 < x and x < 1. The last two inequalities and AxiomOM imply x 2 < x. Since x 0, Proposition 3.6 implies 0 < x 2. Since x 2 < x and x < 1, AxiomOT imply x 2 < 1. Now we have, x 2 Z and 0 < x 2 and x < 1. Thus, x 2 S and also x 2 < x. Hence we have proved that for every x S there exists y = x 2 S such that y = x 2 < x. That is, S does not have a minimum. By 4.3, we deduce S =. The next proposition can be deduced from the previous one. However, I will give a direct proof. Proposition 4.3. The minimum of Z + is 1. Proof. Since 1 Z + the set Z + is not empty. Since clearly Z + Z +, AxiomWO implies that Z + has a minimum. Denote by m the minimum of Z + ; that is, set m = minz +. Recall that m has the following properties: m Z + and x Z + m x. 4.4 Since 1 Z + we have m 1. Since m > 0, Axiom OM yields m 2 m. Since 0 < m by Axiom OM 0 < m 2. Thus m 2 Z +. Since m = minz + we conclude m m 2. Since both m 2 m and m m 2, we have m = m 2, that is mm 1 = 0. Now Axiom MZ implies m = 0 or m 1 = 0. Since m > 0, disjunctive syllogism yields m 1 = 0. That is m = 1 is proved. Definition 4.4. An integer a is a square if there exists an integer b such that a = b 2. Proposition 4.5. Let s be an integer. If s and 2s are both square, then s = 0. Proof. In this proof we shall use the fact that an integer x is even if and only if x 2 is even. Consider the set Clearly S Z +. S = { s Z s > 0, s and 2s are squares }. Next we shall prove that S does not have a minimum. Let s S be arbitrary. Then s > 0 and there exist positive integers j and k such that s = j 2 and 2s = k 2. Since k 2 is even, the integer k is even. Therefore there exist a positive integer m such that k = 2m. Thus, 2s = 4m 2, or, equivalently s = 2m 2. Clearly, m 2 < 2m 2 = s. Since m is positive, m 2 > 0. Now we have that m 2 > 0 and both integers m 2 and 2m 2 = s are square. Therefore m 2 S and m 2 < s. Consequently s is not a minimum of S. Since s S was arbitrary element in S, we have proved that S does not have a minimum. By AxiomWO, see 4.3, 7

8 S =. Thus, there are no positive integers s such that both s and 2s are squares. Therefore, if s and s 2 are both square, then s 0. Since for each square number s by Proposition 3.6 we have s 0, we conclude that s = 0. Proposition 4.6. If p Z and q Z +, then p 2 2q 2. Proof. Let p Z and q Z +. Proposition 4.5 is equivalent to: If s is a square and s 0, then 2s is not a square. Applying this to q 2 we conclude that 2q 2 is not a square. Since p 2 is a square, we conclude p 2 2q 2. The preceding proposition implies that a square of a rational number cannot equal 2. In other words, 2 is irrational. 5 Proof of the principle of mathematical induction In the next theorem the universe of discourse is the set Z + of positive integers. Theorem 5.1. Let Pn be a propositional function involving a positive integer n. Then P1 k Pk Pk+1 n Pn Proof. Recall that the implications p q r and p r q are equivalent. Therefore we will prove: P1 j Pj k Pk Pk Assume P1 and j Pj. That is, assume that there exists j 0 Z + such that Pj 0. Now consider the set S = { n Z + Pn } = { n n Z + Pn }. Clearly S Z + and j 0 S. Hence S Z + S is true. This and AxiomWO, via modus ponens, yield that the set S has a minimum, that is, m S x S m x

9 Since m S we have m Z + and Pm. Since P1 is true we have m > 1. Consequently, by Axiom OA and the definition of 1, we have m 1 > 0. Thus, m 1 Z +. Next we rewrite the proposition 5.2. First, we notice that the proposition x S m x is equivalent to which is further equivalent to Thus 5.2 is equivalent to x x S m x, x x < m x S. m S x x < m x S. 5.3 Define k = m 1. Then k Z +. Further, since k < m, 5.3 implies k S. Since n S is equivalent to n Z + Pn, k S is equivalent to k Z + Pk. Since k Z +, by disjunctive syllogism we deduce Pk is true. Recall that k +1 = m S. Hence Pk +1 is true. Thus, by setting k = m 1, we just proved that k Pk Pk +1 is true. This completes the proof. 6 Even and Odd Integers First recall the definitions of even and odd integers. The set of all even integers we denote by E and the set of all odd integers we denote by O. For n Z we define Proposition 6.1. E O =. n E k Z n = 2k, 6.1 n O k Z n = 2k Proof. We prove this by contradiction. Assume that E O is a nonempty set. Let n E O. Then there exist k,j Z such that n = 2k = 2j+1. Hence, there exists m = k j Z such that 1 = 2m. Recall that we proved 0 < 1 < 2. Substituting 1 = 2m, we get 0 < 2m < 2. Since 0 < 2, Proposition 3.3 applied to 2 0 < 2m, yields 0 < m. Proposition 3.3 applied to 2m < 2 1, yields m < 1. Thus, we have 0 < m and m < 1 and m Z. By Proposition 4.2 the statement 0 < m and m < 1 and m Z is false. Since the assumption that E O leads to a false statement, we proved E O =. 9

10 Proposition 6.2. E O = Z. Proof. First we prove that Z + E O. In other words we prove n Z + n E n O The last displayed statement can be proved by Mathematical Induction. Set Pn to be n E n O. Since 1 = we have 1 O. Therefore, 1 E 1 O is true. So, the base step P1 is true. Next we prove the inductive step. Let n Z + be arbitrary and prove the implication Pn Pn + 1. Assume that Pn is true. That is, assume that n E n O. Consider two cases. For Case 1, assume n E. Clearly n E implies n+1 O. Therefore, n + 1 E n + 1 O is true. Thus Pn+1 holds in this case. For Case 2, assume n O. Clearly n O implies n+1 E. Therefore, n+1 E n+1 O is true. Thus Pn+1 holds in this case as well. Thus, for every n Z + we proved that Pn Pn+1. By Mathematical induction, this proves that n Z + we have n E n O. In other words, Z + E O. Since 0 = 2 0, we have 0 E. Hence, 0 E O. Finally we prove that Z E O. Let n Z be such that n < 0. Then n > 0 and thus, n E O. Now consider two cases. Case 1: if n E, then n = 2k for some k Z. Hence, n = 2 k with k Z. Consequently, n E. Case 2: if n O, then n = 2j + 1 for some j Z. Hence, n = 2 j 1 = 2 j with j 1 Z. Consequently, n O. In either case, n E O. In conclusion, we have proved that for every n Z we have n E O. That is E O = Z. 7 The division algorithm The following theorem is called the division algorithm. Proposition 7.1. Let n be an integer and let d be a positive integer. Then there exist unique integers q and r such that n = dq +r and 0 r < d. Proof. Let n Z and let d Z +. Define the set { S = k Z k 0 j Z k = n dj }. By the definition of S we have S Z and S is bounded below by 0. 10

11 Next we prove that S is a nonempty set. We distinguish two cases for n: n 0 and n < 0. If n 0, then n S since n = n d 0 0. Now assume that n < 0. Then n > 0. Now n > 0 and d 1, imply nd n. Adding n to both sides of nd n we get n dn n n = 0. Since with j = n k = n dj = n dn S we have proved that S in this case. Thus, in each case we identified an integer in S, so S is a nonempty set. Since S is both bounded below and nonempty, Proposition?? implies that S has a minimum. Denote that minimum by r. The integer r has the following two properties: r S and r k for all k S. Since r S, we have r 0 and there exists q Z such that r = n dq. Hence we proved that there exist integers r and q such that n = dq + r and r 0. It remains to prove that r < d. Consider the integer r d. As d > 0 we have r d < r. Since x S implies r x, the contrapositive of the last implication yields r d S. Since x S x 0 j Z x = n dj r d S means r d < 0 j Z r d n dj. 7.1 However, we know that the following is true Thus r d = n dq d = n dq +1. j Z r d = n dj. 7.2 By disjunctive syllogism, 7.1 and 7.2 yield r d < 0. That is r < d. It remains to prove the uniqueness of r and q. Assume that q,r,q,r are integers such that n = dq+r 0 r < d and n = dq +r 0 r < d. Then dq +r = dq +r and 0 r < d and d < r 0. Simplifying the first equality and adding the last two inequalities we get Hence Since 0 < d, Proposition 3.3 implies r r = dq q and d < r r < d. d < dq q < d. 1 < q q < 1. In Proposition 4.2 we proved that there are no integers between 0 and 1. Since q q is an integer we must have 1 < q q 0. Multiplying by 1 < 0 and using Proposition 3.2 we conclude 0 q + q < 1. Now Proposition 4.2 yields q +q = 0. That is q = q. Since r r = dq q we also conclude that r = r. 11

12 Definition 7.2. The integer r in Proposition 7.1 is called the remainder left by n when divided by m. Example 7.3. When divided by 5, the integer 17 leaves a remainders of 2: 17 = When divided by 5, the integer 17 leaves a remainder of 3: 17 = Definition 7.4. Let n be an integer. Let r be the remainder left by n when divided by 2. Then r = 0 or r = 1. We say that n is even if r = 0 and that n is odd if r = 1. Remark 7.5. In Proposition 7.1 we proved that for every n Z and every d Z + there exist unique q Z and unique r Z such that Dividing both relations in 7.3 by d > 0 we get Therefore we have n = dq +r and 0 r < d. 7.3 n d = q + r d and 0 r d < 1. q Z and q n d < q +1. The last displayed line is exactly the definition of the floor of n. Thus, in the division d algorithm n n q = and r = n d d d 12