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Chapter 2: Problem 19

Prove or disprove: If \(m, n\) are positive integers and \(m, n\) are perfect squares, then \(m+n\) is a perfect square.

Short Answer

Expert verified
The statement 'If \(m, n\) are positive integers and \(m, n\) are perfect squares, then \(m+n\) is a perfect square.' is false. A counter example is \(m=4\) and \(n=9\). The sum of \(m\) and \(n\) is 13, which is not a perfect square.

Step by step solution

01

Identify a suitable pair of perfect squares

Remember that a perfect square is an integer that is the square of an integer. For example: \(1^2 = 1, 2^2 = 4, 3^2 = 9, 4^2 = 16\), and so on. The choice of the numbers \(m\) and \(n\) is critical. Since the problem states \(m,n\) are perfect squares, we can choose \(m = 4\) and \(n = 9\) as valid perfect squares.
02

Calculate Sum of the chosen numbers

Next, add the chosen numbers \(m\) and \(n\) together to see if the sum is a perfect square. The sum \(m+n = 4+9 = 13\).
03

Analyze the sum

Check if \(13\) is a perfect square. It is not because \(13\) cannot be expressed as the square of an integer. The squares closest to \(13\) are \(3^2 = 9\n) and \(4^2 = 16\). This contradicts the original statement, proving it incorrect.
04

Conclude

We have found an example that contradicts the statement. Therefore, the given statement is false. Not all sums of perfect squares are perfect squares.

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Key Concepts

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

Integer Properties
When dealing with numbers, it is essential to understand the concept of integer properties. In mathematics, integers include all whole numbers that can be positive, negative, or zero. Perfect squares are a particular subset of integers where a number is the square of another integer. For example, 4 is a perfect square because it equals \(2^2\), and 9 is a perfect square because it equals \(3^2\).

One vital property of integers to remember is they follow basic operations of arithmetic such as addition, subtraction, multiplication, and division. When adding two perfect squares, their sum doesn't inherently result in another perfect square, as demonstrated in the exercise above. Here, the sum of 4 and 9 results in 13, which is not a perfect square.

Understanding the properties of integers, including how they behave under various operations, helps in evaluating mathematical expressions and problem-solving. This groundwork establishes the necessary framework to evaluate statements involving integers, such as determining the truth of conjectures related to their properties and sums.
Mathematical Proofs
In mathematics, proving or disproving a statement relies heavily on logic and established rules. Mathematical proofs are structured arguments that validate the truthfulness of a proposition. A proof involves a logical sequence leading from assumptions (or given information) to the desired conclusion, ensuring each step adheres to rigorous principles.

To disprove a statement like in this exercise, finding a counterexample can effectively demonstrate the proposition's falsehood. In this exercise, the assumption was that the sum of two perfect squares is also a perfect square. However, by choosing specific values, we showed that although both 4 and 9 are perfect squares, their sum, 13, is not. This serves as evidence that the original statement does not hold in all instances, hence proving it wrong through a logical and systematic approach.
Counterexample Method
Sometimes, proving a statement wrong is more straightforward than proving it right; this is where the counterexample method shines. A counterexample is a specific case where the general statement fails to hold true. Demonstrating even one case where the statement is false is sufficient to disprove it.

In the context of the exercise, the conjecture that the sum of any two perfect squares is also a perfect square was challenged by the choice of \(m = 4\) and \(n = 9\). Their sum, 13, served as a counterexample since 13 isn't a perfect square. Since a single valid counterexample contravenes the statement, it is disproved immediately.

Using counterexamples is a powerful and efficient technique in mathematical proofs, allowing us to dismiss incorrect conjectures quickly. This method focuses only on finding an instance where the rule doesn鈥檛 apply, rather than evaluating its validity across all possible scenarios.

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Most popular questions from this chapter

Let \(p(x), q(x)\), and \(r(x)\) be the following open statements. $$ \begin{array}{ll} p(x): & x^{2}-7 x+10=0 \\ q(x): & x^{2}-2 x-3=0 \\ r(x): & x<0 \end{array} $$a) Determine the truth or falsity of the following statements, where the universe is all integers. If a statement is false, provide a counterexample or explanation. i) \(\forall x[p(x) \rightarrow \neg r(x)]\) ii) \(\forall x[q(x) \rightarrow r(x)]\) iii) \(\exists x[q(x) \rightarrow r(x)]\) iv) \(\exists x[p(x) \rightarrow r(x)]\) b) Find the answers to part (a) when the universe consists of all positive integers. c) Find the answers to part (a) when the universe contains only the integers 2 and \(5 .\)

Let \(p, q, r\) denote primitive statements. Write the converse, inverse, and contrapositive of each of the following implications. a) \(p \rightarrow(q \wedge r)\) b) \((p \vee q) \rightarrow r\)

Provide the reasons for the steps verifying the following argument. (Here a denotes a specific but arbitrarily chosen element from the given universe.) $$ \begin{aligned} &\forall x[p(x) \rightarrow(q(x) \wedge r(x))] \\ &\frac{\forall x[p(x) \wedge s(x)]}{\therefore \forall x[r(x) \wedge s(x)]} \end{aligned} $$ Steps Rensons 1) \(\forall x[p(x) \rightarrow(q(x) \wedge r(x))]\) 2) \(\forall x[p(x) \wedge s(x)]\) 3) \(p(a) \rightarrow(q(a) \wedge r(a))\) 4) \(p(a) \wedge s(a)\) 5) \(p(a)\) 6) \(q(a) \wedge r(a)\) 7) \(r(a)\) 8) \(s(a)\) 9) \(r(a) \wedge s(a)\) 10) \(\therefore \forall x[r(x) \wedge s(x)]\)

For primitive statements \(p, q\), a) verify that \(p \rightarrow[q \rightarrow(p \wedge q)]\) is a tautology. b) verify that \((p \vee q) \rightarrow[q \rightarrow q]\) is a tautology by using the result from part (a) along with the substitution rules and the laws of logic. c) is \((p \vee q) \rightarrow[q \rightarrow(p \wedge q)]\) a tautology?

Express the negation of the statement \(p \leftrightarrow q\) in terms of the connectives \(\wedge\) and \(V\)
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