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Chapter 1: Problem 3

Show that the square of an even number is an even number using a direct proof.

Short Answer

Expert verified
The square of any even number \( n = 2k \) is always even, as \( n^2 = 4k^2 \) which is divisible by 2.

Step by step solution

01

Define an even number

An even number can be expressed as any integer that is divisible by 2. Let's represent an even number as \( n = 2k \) where \( k \) is an integer.
02

Square the even number

To show that the square of an even number is also even, we first square the expression for an even number. The expression is \( n = 2k \). So, squaring both sides we get \( n^2 = (2k)^2 \).
03

Simplify the squared number

Simplify \( (2k)^2 \). It can be rewritten as \( (2k) \times (2k) = 4k^2 \).
04

Factor the result

Notice that \( 4k^2 \) can be factored to \( 2(2k^2) \). This shows that \( 4k^2 \) is clearly divisible by 2, indicating that \( 4k^2 \) is an even number.
05

Conclude the proof

Since \( n^2 = 4k^2 \) is an even number, we have shown that the square of an even number is indeed even.

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Key Concepts

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

Even Numbers
An even number is any integer that can be exactly divided by 2. To put it simply, even numbers are those numbers that end in 0, 2, 4, 6, or 8 when written in decimal form. Some examples include 2, 4, 6, 8, 10, and so on. If you express an even number algebraically, it can be written as \( n = 2k \), where \( k \) is an integer. This formula helps us to understand and manipulate even numbers in mathematical proofs and problems. By knowing this, we can easily identify and work with even numbers in various calculations and proofs.
Mathematical Proof
Mathematical proof is a logical series of statements leading from the assumptions to the conclusion. Direct proof is a type of mathematical proof. In direct proof, you start with what is given and use logical steps to arrive at the statement that needs to be proven.
Steps used in direct proof:
  • 1. State what is given and what needs to be proven.
  • 2. Introduce definitions relevant to the problem (e.g., even number).
  • 3. Use algebraic manipulations to show the statement is true.
  • 4. Conclude by stating that the original claim has been proven.
Using direct proofs can make showing mathematical concepts and relationships clear and straightforward.
Squaring Integers
Squaring an integer means multiplying the integer by itself. For example, if we square the integer 3, we get \( 3^2 = 9 \). When it comes to even numbers, we can use the expression for an even number, \( n = 2k \), to illustrate the concept. If we square an even number, like so:
\[ (2k)^2 = 4k^2 \]
We see that the result, \( 4k^2 \), is also a multiple of 2 (since \( 4 = 2 \times 2 \)). Therefore, \( 4k^2 \) is another even number. This proof shows that the square of an even number always results in another even number. Understanding this concept helps in building a solid foundation for more advanced mathematical operations and theorems.

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Most popular questions from this chapter

Use quantifiers and logical connectives to express the fact that a quadratic polynomial with real number coefficients has at most two real roots.

A statement is in prenex normal form (PNF) if and only if it is of the form $$ Q_{1} x_{1} Q_{2} x_{2} \cdots Q_{k} x_{k} P\left(x_{1}, x_{2}, \ldots, x_{k}\right) $$ where each \(Q_{i}, i=1,2, \ldots, k,\) is either the existential quantifier or the universal quantifier, and \(P\left(x_{1}, \ldots, x_{k}\right)\) is a predicate involving no quantifiers. For example, \(\exists x \forall y(P(x, y) \wedge Q(y))\) is in prenex normal form, whereas \(\exists x P(x) \vee \forall x Q(x)\) is not (because the quantifiers do not all occur first). Every statement formed from propositional variables, predicates, \(\mathbf{T},\) and \(\mathbf{F}\) using logical connectives and quantifiers is equivalent to a statement in prenex normal form. Exercise 51 asks for a proof of this fact. Show how to transform an arbitrary statement to a statement in prenex normal form that is equivalent to the given statement. (Note: A formal solution of this exercise requires use of structural induction, covered in Section \(5.3 . )\)

Exercises \(40-44\) deal with the translation between system specification and logical expressions involving quantifiers. Express each of these system specifications using predicates, quantifiers, and logical connectives. a) Every user has access to an electronic mailbox. b) The system mailbox can be accessed by everyone in the group if the file system is locked. c) The firewall is in a diagnostic state only if the proxy server is in a diagnostic state. d) At least one router is functioning normally if the throughput is between 100 kbps and 500 kbps and the proxy server is not in diagnostic mode.

Suppose the domain of the propositional function \(P(x, y)\) consists of pairs \(x\) and \(y,\) where \(x\) is \(1,2,\) or 3 and \(y\) is \(1,2,\) or \(3 .\) Write out these propositions using disjunctions and conjunctions. $$ \begin{array}{ll}{\text { a) } \forall x \forall y P(x, y)} & {\text { b) } \exists x \exists y P(x, y)} \\ {\text { c) } \exists x \forall y P(x, y)} & {\text { d) } \forall y \exists x P(x, y)}\end{array} $$

Let M(x, y) be 鈥渪 has sent y an e-mail message鈥� and T(x, y) be 鈥渪 has telephoned y,鈥� where the domain consists of all students in your class. Use quantifiers to express each of these statements. (Assume that all e-mail messages that were sent are received, which is not the way things often work.) a) Chou has never sent an e-mail message to Koko. b) Arlene has never sent an e-mail message to or tele- phoned Sarah. c) Jose has never received an e-mail message from Deborah. d) Every student in your class has sent an e-mail mes- sage to Ken. e) No one in your class has telephoned Nina. f ) Everyone in your class has either telephoned Avi or sent him an e-mail message. g) There is a student in your class who has sent every- one else in your class an e-mail message. h) There is someone in your class who has either sent an e-mail message or telephoned everyone else in your class. i) There are two different students in your class who have sent each other e-mail messages. j) There is a student who has sent himself or herself an e-mail message. k) There is a student in your class who has not received an e-mail message from anyone else in the class and who has not been called by any other student in the class. l) Every student in the class has either received an email message or received a telephone call from another student in the class. m) There are at least two students in your class such that one student has sent the other e-mail and the second student has telephoned the first student. n) There are two different students in your class who between them have sent an e-mail message to or telephoned everyone else in the class.
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