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Chapter 1: Problem 30

Prove that \(m^{2}=n^{2}\) if and only if \(m=n\) or \(m=-n\)

Short Answer

Expert verified
If \(m^2 = n^2\), then \(m = n\) or \(m = -n\).

Step by step solution

01

Understand the Problem

The goal is to prove that if the squares of two numbers \(m\) and \(n\) are equal, then either \(m\) is equal to \(n\) or \(m\) is the opposite of \(n\) (\(m = -n\)).
02

Express the Given Equation

Start with the equation given in the problem: \[ m^2 = n^2 \]
03

Factorize the Equation

Write the equation in a factored form to simplify it: \[ m^2 - n^2 = 0 \] \[ (m - n)(m + n) = 0 \]
04

Apply the Zero Product Property

If the product of two factors is zero, then at least one of the factors must be zero. So, set each factor to zero and solve for \(m\):\( m - n = 0 \) \(m = n \)or\( m + n = 0 \) \(m = -n \)
05

Combine the Results

From the previous step, it follows that the equation \( m^2 = n^2 \) holds true if and only if \( m = n \) or \( m = -n \).
06

Write the Conclusion

Thus, it is proven that \( m^2 = n^2 \) if and only if \( m = n \) or \( m = -n \).

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Key Concepts

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

Equation Factorization
The goal of equation factorization is to simplify complex equations, making them easier to solve. It's like breaking down a difficult problem into smaller, more manageable pieces. In our exercise, we start with the quadratic equation:
(m^2 = n^2)
Factorizing this equation involves rewriting it in a form that includes multiplication. Here's how we do it:
Susbtract (n^2) from both sides:
(m^2 - n^2 = 0)
Notice that the left side of the equation is a difference of squares. So, we factorize:
(m - n)(m + n) = 0

Factorization helps us move to the next step, allowing us to apply other key concepts.
Zero Product Property
The Zero Product Property is a powerful tool in algebra. It states that if the product of two numbers (or expressions) is zero, then at least one of those numbers must be zero. In our problem, after factorizing, we have:
(m - n)(m + n) = 0

According to the Zero Product Property, this implies:
  • m - n = 0 or

  • m + n = 0


Solving these equations gives us:
  • m - n = 0 implies m = n

  • m + n = 0 implies m = -n


The importance of the Zero Product Property helps us divide the problem into simpler solutions, making the logical proof possible.
If and Only If Proof
An 'if and only if' proof, often abbreviated as 'iff', establishes that two statements are logically equivalent. That means each statement is both necessary and sufficient for the other. In our case, we need to show that:
m^2 = n^2 if and only if m = n or m = -n

The steps we followed earlier accomplish this:

  • We started with m^2 = n^2.

  • We rewrote it using factorization: (m - n)(m + n) = 0

  • We applied the Zero Product Property to find: m = n or m = -n



This logical sequence works both ways, proving the 'if and only if' relationship. This concept ensures clarity and confirms the equivalence between the conditions.

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Most popular questions from this chapter

Determine whether each of these arguments is valid. If an argument is correct, what rule of inference is being used? If it is not, what logical error occurs? a) If \(n\) is a real number such that \(n>1,\) then \(n^{2}>1\) Suppose that \(n^{2}>1 .\) Then \(n>1\) b) If \(n\) is a real number with \(n>3,\) then \(n^{2}>9\) . Suppose that \(n^{2} \leq 9 .\) Then \(n \leq 3\) . c) If \(n\) is a real number with \(n>2,\) then \(n^{2}>4\) . Suppose that \(n \leq 2 .\) Then \(n^{2} \leq 4 .\)

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. $$ \begin{array}{ll}{\text { a) } \forall x \forall y P(x, y)} & {\text { b) } \exists x \exists y P(x, y)} \\ {\text { c) } \exists x \forall y P(x, y)} & {\text { d) } \forall y \exists x P(x, y)}\end{array} $$

Use rules of inference to show that if \(\forall x(P(x) \vee Q(x))\) \(\forall x(\neg Q(x) \vee S(x)), \quad \forall x(R(x) \rightarrow \neg S(x)),\) and \(\exists x \neg P(x)\) are true, then \(\exists x \neg R(x)\) is true.

Fuzzy logic is used in artificial intelligence. In fuzzy logic, a proposition has a truth value that is a number between 0 and 1, inclusive. A proposition with a truth value of 0 is false and one with a truth value of 1 is true. Truth values that are between 0 and 1 indicate varying degrees of truth. For instance, the truth value 0.8 can be assigned to the statement 鈥淔red is happy,鈥� because Fred is happy most of the time, and the truth value 0.4 can be assigned to the statement 鈥淛ohn is happy,鈥� because John is happy slightly less than half the time. Use these truth values to solve The truth value of the negation of a proposition in fuzzy logic is 1 minus the truth value of the proposition. What are the truth values of the statements 鈥淔red is not happy鈥� and 鈥淛ohn is not happy鈥�?

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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