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Problem 0.3. Prove the following (you may use that the sum and product of natural numbers are natural numbers) (1) If a < b then for any e we have a + c < b + c (2) If a < b andb

Problem 0.3. Prove the following (you may use that the sum and product of natural numbers are natural numbers) (1) If a < b then for any e we have a + c < b + c (2) If a < b andb

Advanced Engineering Mathematics
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
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problem 0.3

Just the basic arithmetic rules aren't enough to prove everything we want to know about the integers. For example, showing that 2 does not divide 3 means showing that there is no integer \( x \) with \( 2x = 3 \). Notice that there are solutions in the reals or rationals, so to prove what we want we need something that is true for the integers but not for those other kinds of numbers. Another way to think of this is to say that we need to define more carefully what we mean by an integer. This is surprisingly difficult, and before we can do that we need to develop some more tools. The next piece that we want to incorporate is that we can compare integers to see which is bigger. This is what mathematicians call an ordering. To be more precise: **Definition 0.1.** An integer \( n \) is positive if and only if \( n \in \mathbb{N} \) **Definition 0.2.** For any two integers \( a \) and \( b \), we say \( a < b \) if and only if \( b - a \) is positive. We can now prove many of the basic facts we know about comparing integers. **Problem 0.3.** *Prove the following (you may use that the sum and product of natural numbers are natural numbers)* 1. If \( a < b \) then for any \( c \) we have \( a + c < b + c \) 2. If \( a < b \) and \( b < c \) then \( a < c \) 3. For any \( a \) and \( b \) exactly one of the following holds: \( a < b, b < a, \text{ or } a = b \) 4. If \( a \) is positive and \( b \) is negative then \( ab \) is negative 5. If \( a \) is negative and \( b \) is negative then \( ab \) is positive 6. Let \( a \) and \( b \) be integers with \( a < b \), then for all \( c > 0 \) we have \( ac < bc \) and for all \( c < 0 \) we have \( bc < ac \) (be careful, the assumption that \( c > 0 \) is essential so your proof better use it
Transcribed Image Text:Just the basic arithmetic rules aren't enough to prove everything we want to know about the integers. For example, showing that 2 does not divide 3 means showing that there is no integer \( x \) with \( 2x = 3 \). Notice that there are solutions in the reals or rationals, so to prove what we want we need something that is true for the integers but not for those other kinds of numbers. Another way to think of this is to say that we need to define more carefully what we mean by an integer. This is surprisingly difficult, and before we can do that we need to develop some more tools. The next piece that we want to incorporate is that we can compare integers to see which is bigger. This is what mathematicians call an ordering. To be more precise: **Definition 0.1.** An integer \( n \) is positive if and only if \( n \in \mathbb{N} \) **Definition 0.2.** For any two integers \( a \) and \( b \), we say \( a < b \) if and only if \( b - a \) is positive. We can now prove many of the basic facts we know about comparing integers. **Problem 0.3.** *Prove the following (you may use that the sum and product of natural numbers are natural numbers)* 1. If \( a < b \) then for any \( c \) we have \( a + c < b + c \) 2. If \( a < b \) and \( b < c \) then \( a < c \) 3. For any \( a \) and \( b \) exactly one of the following holds: \( a < b, b < a, \text{ or } a = b \) 4. If \( a \) is positive and \( b \) is negative then \( ab \) is negative 5. If \( a \) is negative and \( b \) is negative then \( ab \) is positive 6. Let \( a \) and \( b \) be integers with \( a < b \), then for all \( c > 0 \) we have \( ac < bc \) and for all \( c < 0 \) we have \( bc < ac \) (be careful, the assumption that \( c > 0 \) is essential so your proof better use it
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