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Chapter 2: Problem 16

16\. Prove that for every integer \(n, n^{2}\) is even if and only if \(n\) is even.

Short Answer

Expert verified
The statement is proven to be true. For every integer \(n\), \(n^{2}\) is even if and only if \(n\) is even.

Step by step solution

01

Formulate Assumptions

Assume that \(n\) is an integer and \(n^{2}\) is even. By the properties of even numbers, this means that \(n^{2}\) can be expressed as \(2k\) for some integer \(k\).
02

Express n in terms of other Integers

Now express \(n\) as \(2l\) or \(2l+1\) where \(l\) is also an integer. This covers all possible integers.
03

Substitute n in terms of l into the expression for n squared

Substitute \(n = 2l\) or \(n = 2l+1\) into the expression for \(n^{2}\) to see how it affects the evenness or oddness of the result.
04

Prove that if n squared is even, then n is even

When you work out \(n^{2} = (2l)^{2} = 4l^{2} = 2(2l^{2})\), you can see that the resulting number is even since it could be written as \(2m\) where \(m = 2l^{2}\) is an integer. This means that if \(n^{2}\) is even, then \(n\) has to be even.
05

Prove that if n is even, then n squared is even

Now consider if \(n = 2l\). Then \(n^{2} = (2l)^{2} = 4l^{2} = 2(2l^{2})\). Again, this number is even since it can be written as \(2m\) where \(m = 2l^{2}\) is an integer. This means that, if \(n\) is even, then \(n^{2}\) must also be even.
06

Conclude Proof

Together, steps 4 and 5 complete the proof. It was shown that if \(n^{2}\) is even, then \(n\) is also even, and inversely, if \(n\) is even, then \(n^{2}\) is also even. This has covered all possible cases, so the statement has been proven.

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Key Concepts

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

Even and Odd Integers
In mathematics, numbers can often be categorized as either even or odd. This distinction is simple but powerful when understanding number properties. Even integers are those that can be divided by 2 without a remainder. Examples include 2, 4, 6, etc. Mathematically, we express an even integer as \( n = 2k \), where \( k \) is an integer.

Odd integers, conversely, leave a remainder of 1 when divided by 2. Classic examples are 1, 3, 5, etc. We can express these as \( n = 2k + 1 \). Recognizing whether a number is even or odd is essential in many proofs because their products and sums also follow predictable patterns.

Understanding the nature of integers as even or odd helps simplify complex calculations, making problem solving more manageable.
Properties of Integers
Integers possess various properties that make them unique and helpful when forming proofs and solving equations. A crucial property is their closure under addition, subtraction, and multiplication. This means combining any two integers with these operations always results in another integer.

Integers also exhibit commutative and associative properties over addition and multiplication. For example, for any integers \( a \) and \( b \), \( a + b = b + a \) is true (commutative), and \((a + b) + c = a + (b + c)\) is true for any integer \( c \) (associative).

Another critical aspect is the property of zero and one. Zero, as an integer, is the identity element for addition, while one is the identity element for multiplication. These properties ensure that calculations and transformations involving integers remain consistent and predictable.
Integer Expressions
Integer expressions involve using integers within mathematical expressions, often making use of arithmetic operations like addition, subtraction, multiplication, and division. Understanding how to manipulate these expressions is crucial for solving equations and proving mathematical statements.

Consider \( n^{2} \) in expressions. When exploring products or powers, knowing whether \( n \) is even or odd can simplify the expression. For example, if \( n \) is even, then \( n = 2k \), and \( n^{2} = (2k)^{2} = 4k^{2} \), clearly showing that \( n^{2} \) is even.

Expressions become even more insightful when we substitute known properties of integers into them. For instance, using the distributive property \( a(b + c) = ab + ac \), can break down complex integer expressions into more manageable parts for easier calculation and understanding.

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Most popular questions from this chapter

6\. Determine the truth value of cach of the following impliations. a) If \(3+4=12\), then \(3+2=6\) b) If \(3+3=6\), then \(3+4=9\) c) If Thomas Jefferson was the third president of the United States, then \(2+3=5\).

8\. Give the reasons for the steps verifying the following argument. \((\neg p \vee q) \rightarrow r\) \(r \rightarrow(s \vee t)\) \(\quad \neg s \wedge \neg u\) \(\frac{\neg u \rightarrow \neg t} \\\\{\therefore p} \end{array\) Steps Reasons 1) \(\neg s \wedge \neg u\) 2) \(\neg u\) 3) \(\neg u \rightarrow \neg t\) 4) \(\neg t\) 5) \(\neg s\) 6) \(\neg s \wedge \neg t\) 7) \(r \rightarrow(s \vee t)\) 8) \(\neg(s \vee t) \rightarrow \neg r\) 9) \((\neg s \wedge \neg t) \rightarrow \neg r\) 10) \(\neg r\) 11) \((\neg p \vee q) \rightarrow r\) 12) \(\neg r \rightarrow \neg(\neg p \vee q)\) 13) \(\neg r \rightarrow(p \wedge \neg q)\) 14) \(p \wedge \neg q\) 15) \(\therefore p\)

14\. At the start of a program (written in pseudocode) the integer variable \(n\) is assigned the value \(7 .\) Determine the value of \(n\) after each of the following successive statements is encountered during the execution of this program. [Here the value of \(n\) following the execution of the statement in part (a) becomes the value of \(n\) for the statement in part (b), and so on, through the statement in part (d). For positive integers \(a, b,\lfloor a / b\rfloor\) returns the integer part of the quotient - for example, \(\lfloor 6 / 2\rfloor=3\), \(\lfloor 7 / 2\rfloor=3,\lfloor 2 / 5\rfloor=0\), and \([8 / 3\rfloor=2 .]\) a) if \(n>5\) then \(n:=n+2\) b) if \(((n+2=8)\) or \((n-3=6)\) ) then \(n:=2 * n+1\) c) if \(((n-3=16)\) and \((\lfloor n / 6\rfloor=1)\) ) then \(n:=n+3\) d) if \((n \neq 21)\) and \((n-7=15)\) ) then \(n:=n-4\)

8\. Construct a truth table for each of the following compound statements, where \(p, q, r\) denote primitive statements. a) \(\neg(p \vee \neg q) \rightarrow \neg p\) b) \(p \rightarrow(q \rightarrow r)\) c) \((p \rightarrow q) \rightarrow r\) d) \((p \rightarrow q) \rightarrow(q \rightarrow p)\) e) \([p \wedge(p \rightarrow q)] \rightarrow q\) f) \((p \wedge q) \rightarrow p\) g) \(q \leftrightarrow(\neg p \vee \neg q)\) h) \([(p \rightarrow q) \wedge(q \rightarrow r)] \rightarrow(p \rightarrow r)\)

4\. Let \(p, q, r, s\) denote the following statements: \(p\) : I finish writing my computer program before lunch. q: I shall play tennis in the afternoon. \(r\) : The sun is shining. s: The humidity is low. Write the following in symbolic form. a) If the sun is shining. I shall play tennis this afternoon. b) Finishing the writing of my computer program before lunch is necessary for my playing tennis this afternoon. c) Low humidity and sunshine are sufficient for me to play tennis this afternoon.
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