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Chapter 6: Problem 12

Prove or disprove the following: (a) The sum of two even integers is an even integer. (b) The sum of two odd integers is an odd integer (c) The sum of two primes is never a prime. (d) The sum of three consecutive integers is divisible by 3 Indicate the methods of proof where appropriate.

Short Answer

Expert verified
(a) True; (b) False; (c) False; (d) True.

Step by step solution

01

Understanding Even Integers

An even integer is one that can be expressed as \( 2k \), where \( k \) is an integer.
02

Prove the Sum of Two Even Integers (a)

Let \( a = 2m \) and \( b = 2n \), where \( m \) and \( n \) are integers. Their sum is \( a + b = 2m + 2n = 2(m + n) \). Since \( 2(m+n) \) is of the form \( 2k \), the sum is even. This is a direct proof.
03

Understanding Odd Integers

An odd integer is one that can be expressed as \( 2k + 1 \), where \( k \) is an integer.
04

Disprove (b) The Sum of Two Odd Integers

Let \( c = 2p + 1 \) and \( d = 2q + 1 \), where \( p \) and \( q \) are integers. Their sum is \( c + d = (2p + 1) + (2q + 1) = 2p + 2q + 2 = 2(p + q + 1) \). Thus, the sum is even, disproving the statement that the sum of two odd integers is odd.
05

Understanding Prime Numbers

A prime number is an integer greater than 1 with no positive divisors other than 1 and itself.
06

Disproving (c) Statement Using Counterexample

Consider the prime numbers 2 and 3; their sum is 5, which is a prime. This counterexample disproves the statement that the sum of two primes is never a prime.
07

Understanding Consecutive Integers

Three consecutive integers can be expressed as \( n, n+1, n+2 \), where \( n \) is an integer.
08

Proof for (d) Using Algebraic Representation

The sum of three consecutive integers \( n, n+1, n+2 \) is \( n + (n+1) + (n+2) = 3n + 3 = 3(n + 1) \). Since \( 3(n + 1) \) is clearly divisible by 3, the statement is proven using an algebraic proof.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Even Integers
Even integers are numbers that can be divided by 2 without leaving a remainder. A number is even if it can be expressed in the form of \( 2k \), where \( k \) is an integer. Consider numbers like 2, 4, 6, and so on鈥攖heir common characteristic is that you can split them into two equal parts.
For example:
  • When you take the number 8, you can think of it as \( 2 \times 4 \). This confirms that 8 is even.
  • The number 14 can be represented as \( 2 \times 7 \), which is also even.
The sum of two even integers is always even, because if you add \( 2m \) and \( 2n \) (both even), the result is \( 2(m+n) \), which follows the form \( 2k \), thereby confirming the sum is even.
Odd Integers
An odd integer is a number that cannot be divided by 2 evenly. These numbers are in the form \( 2k + 1 \), where \( k \) is an integer. This can be numbers like 1, 3, 5, etc. Each of these numbers has a remainder of 1 when divided by 2.
Consider:
  • The number 9 can be written as \( 2 \times 4 + 1 \), hence it is odd.
  • The number 15 fits the equation \( 2 \times 7 + 1 \).
When you sum two odd integers, for example \( 2p + 1 \) and \( 2q + 1 \), the result is \( 2(p+q+1) \). This expression is equivalent to twice an integer \( 2k \), proving that the sum is actually even, which disproves the belief that two odd numbers add up to an odd number.
Prime Numbers
Prime numbers are special. They have exactly two distinct natural number divisors: 1 and the number itself. Examples include 2, 3, 5, and 7. The unique property of prime numbers is a core concept of number theory, used in various applications such as cryptography.
It's important to note:
  • The number 2 is the only even prime number.
  • All other prime numbers are odd.
The assumption that the sum of two primes cannot be a prime is incorrect. A quick example is 2 and 3, whose sum is 5鈥攁lso a prime. This simple counterexample is enough to refute the statement.
Consecutive Integers
Consecutive integers are numbers that follow each other in order without any gaps. A clear way to think about them is as \( n, n+1, n+2 \), where \( n \) is any given integer.
An example:
  • If you take 4 as \( n \), the consecutive integers are 4, 5, and 6.
  • When choosing 7, you have 7, 8, and 9 as consecutive integers.
The sum of three consecutive integers is \( n + (n+1) + (n+2) = 3n + 3 \), which can be factored into \( 3(n+1) \). Since this expression is a multiple of 3, this sums up the fact that it is always divisible by 3, consistently confirming the initial statement.

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Most popular questions from this chapter

(a) Draw up truth tables to represent the statements (i) \(p\) is equivalent to \(q\) (ii) \(p\) implies \(q\) (b) Using the algebra of statements, represent the truth of the statements below in tabular form and hence determine whether they are true or false: (i) If \(p\) implies \(q\), and \(r\) implies \(q\), then either \(r\). implies \(p\) or \(p\) implies \(r\). (ii) If \(p\) is equivalent to \(q\), and \(q\) is equivalent to \(r\), then \(p\) implies \(r\).

Let \(A, B\) and \(C\) be the following propositions: \(A:\) It is raining B: The sun is shining \(C\) : There are clouds in the sky Translate the following into logical notation: (a) It is raining and the sun is shining. (b) If it is raining then there are clouds in the sky. (c) If it is not raining then the sun is not shining and there are clouds in the sky. (d) If there are no clouds in the sky then the sun is shining.

State the converse and contrapositive of each of the following statements: (a) If the train is late, I will not go. (b) If you have enough money, you will retire. (c) I cannot do it unless you are there too. (d) If you go, so will 1 .

Which of the following statements are propositions? For those that are not, say why and suggest ways of changing them so that they become propositions. For those that are, comment on their truth value. (a) Julius Caesar was prime minister of Great Britain. (b) Stop hitting me. (c) Turn right at the next roundabout. (d) The Moon is made of green cheese. (e) If the world is flat then \(3+3=6\). (f) If you get a degree then you will be rich. (g) \(x+y+z=0\). (h) The 140 th decimal digit in the representation of \(\pi\) is \(8 .\) (i) There are five Platonic solids.

Negate the following propositions: (a) Fred is my brother. (b) 12 is an even number. (c) There will be gales next winter. (d) Bridges collapse when design loads are exceeded.
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