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Chapter 1: Problem 1

Which of the following sentences are statements? (a) \(3^{2}+4^{2}=5^{2}\) (b) \(a^{2}+b^{2}=c^{2}\). (c) There exist integers \(a, b,\) and \(c\) such that \(a^{2}=b^{2}+c^{2}\). (d) If \(x^{2}=4,\) then \(x=2\). (e) For each real number \(x,\) if \(x^{2}=4,\) then \(x=2\). (f) For each real number \(t, \sin ^{2} t+\cos ^{2} t=1\). (g) \(\sin x<\sin \left(\frac{\pi}{4}\right)\). (h) If \(n\) is a prime number, then \(n^{2}\) has three positive factors. (i) \(1+\tan ^{2} \theta=\sec ^{2} \theta\) (j) Every rectangle is a parallelogram. (k) Every even natural number greater than or equal to 4 is the sum of two prime numbers.

Short Answer

Expert verified
The sentences that are statements are: (a) \(3^2 + 4^2 = 5^2\) (c) There exist integers \(a, b,\) and \(c\) such that \(a^2 = b^2 + c^2\) (d) If \(x^2 = 4,\) then \(x = 2\) (f) For each real number \(t, \sin^2 t + \cos^2 t = 1\) (h) If \(n\) is a prime number, then \(n^2\) has three positive factors (i) \(1 + \tan^2\theta = \sec^2\theta\) (j) Every rectangle is a parallelogram (k) Every even natural number greater than or equal to 4 is the sum of two prime numbers.

Step by step solution

01

(a)

We need to check if the equation \(3^2 + 4^2 = 5^2\) holds true. Evaluating each side, we find \(3^2 + 4^2 = 9 + 16 = 25\), and \(5^2 = 25\). Since both sides are equal, the equation is a true statement.
02

(b)

In this case, we have a general equation \(a^2 + b^2 = c^2\) without specified values for \(a, b,\) and \(c\). Since this equation can be either true or false for different values of \(a, b,\) and \(c\), it is not a statement.
03

(c)

The sentence claims the existence of integers \(a, b,\) and \(c\) satisfying \(a^2 = b^2 + c^2\). This is a statement, because either there exist such integers or there don't. A specific example of such integers is when \(a = 5, b = 3,\) and \(c = 4\), as we saw in (a).
04

(d)

We are given the condition "If \(x^2 = 4\), then \(x = 2\)". However, we know that \(x\) can also be \(-2\). Since the given condition is false, it is a statement but a false one.
05

(e)

This sentence says "For each real number \(x,\) if \(x^2 = 4\) then \(x = 2\)." This is not a statement, because it is not true for all real numbers. Specifically, \(x = -2\) is a counterexample.
06

(f)

The sentence claims that for each real number \(t, \sin^2 t + \cos^2 t = 1\). This is a statement, known as the Pythagorean identity, which holds true for all real numbers.
07

(g)

We have an inequality \(\sin x < \sin\left(\frac{\pi}{4}\right)\). Since this inequality can be either true or false for different values of \(x\), it is not a statement.
08

(h)

The sentence says "If \(n\) is a prime number, then \(n^2\) has three positive factors." This is a statement, because either the condition is true for all prime numbers or it isn't. In this case, the statement is true, since for a prime number \(n\) of the form \(n=p\), the three factors of \(n^2\) are \(1, p, p^2\).
09

(i)

The sentence claims that \(1 + \tan^2\theta = \sec^2\theta\). This is a statement known as the tangent-secant identity, which holds true for all valid angles \(\theta\).
10

(j)

The sentence says "Every rectangle is a parallelogram." This is a statement and a true definition, since all rectangles have parallel opposite sides, fulfilling the conditions for a parallelogram.
11

(k)

The sentence claims "Every even natural number greater than or equal to 4 is the sum of two prime numbers." This is known as Goldbach's Conjecture and is a statement. Although it hasn't been proven or refuted, it must be one or the other as it is a true or false statement.

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Key Concepts

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

Statements in Mathematics
In mathematics, statements are sentences that are either true or false, but not both. These are essential for forming the backbone of mathematical reasoning. A statement should be definite, meaning it can be tested for validity. Here are some examples to understand this concept better:
  • True Statement: An example is (a) \(3^2 + 4^2 = 5^2\). Both sides are equal, making this a true mathematical statement.
  • Ambiguous Sentence: The sentence (b) \(a^2 + b^2 = c^2\) is ambiguous without additional information. Without specific values for \(a, b,\) and \(c\), this can't be classified as true or false.
  • Conditional Statement: Take (d) "If \(x^2 = 4\), then \(x = 2\)." Although technically a statement, it's based on a false premise since \(x\) could also be \(-2\).
Understanding whether a sentence is a statement helps in determining its role and truthfulness in problem-solving.
Logical Reasoning
Logical reasoning is how we connect statements to derive conclusions in mathematics. It's the process that allows us to assess the truth of complex propositions based on given truths. Mathematical logic often involves conditional statements, where the truth of one statement depends on another.
  • Conditionals: Statements involving 'if... then...' conditions, such as (h) "If \(n\) is a prime number, then \(n^2\) has three positive factors." This is a verifiable statement, showing the power of logical reasoning in determining truth values.
  • Existential Quantifiers: Sentence (c) "There exist integers \(a, b, \) and \(c \) such that \(a^2 = b^2 + c^2\)" is an example involving existence. Logical reasoning can prove or refute such claims by finding examples or counterexamples.
Logical reasoning is essential in mathematics for developing robust arguments and conclusions.
Mathematical Proof
Mathematical proof is the rigorous process of demonstrating that a statement is true beyond any doubt. Proofs rely on logical reasoning and accepted mathematical principles. Here are ways to understand proofs in context:
  • Direct Proof: Uses direct reasoning from known truths to conclude, such as proving the Pythagorean identity \(\sin^2 t + \cos^2 t = 1\) for all \(t\), a statement supported by fundamental trigonometric truths.

  • Counterexample: Used to refute a statement, like refuting (e) "For each real number \(x\), if \(x^2 = 4\) then \(x = 2\)" by noting that \(x = -2\) satisfies \(x^2 = 4\) but does not satisfy \(x = 2\).
  • Conjectures: These are statements believed to be true without proof, such as (k) Goldbach's Conjecture. Mathematical research may involve attempts to prove or disprove such conjectures.
Mathematical proofs are the foundation of establishing truth within the discipline.

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Most popular questions from this chapter

Are the following statements true or false? Justify your conclusions. (a) If \(a, b\) and \(c\) are integers, then \(a b+a c\) is an even integer. (b) If \(b\) and \(c\) are odd integers and \(a\) is an integer, then \(a b+a c\) is an even integer.

Identify the hypothesis and the conclusion for each of the following conditional statements. (a) If \(n\) is a prime number, then \(n^{2}\) has three positive factors. (b) If \(a\) is an irrational number and \(b\) is an irrational number, then \(a \cdot b\) is an irrational number. (c) If \(p\) is a prime number, then \(p=2\) or \(p\) is an odd number. (d) If \(p\) is a prime number and \(p \neq 2,\) then \(p\) is an odd number. (e) If \(p \neq 2\) and \(p\) is an even number, then \(p\) is not prime.

Following is a statement of a theorem which can be proven using the quadratic formula. For this theorem, \(a, b,\) and \(c\) are real numbers. Theorem If \(f\) is a quadratic function of the form \(f(x)=a x^{2}+b x+c\) and \(a c<0,\) then the function \(f\) has two \(x\) -intercepts. Using only this theorem, what can be concluded about the functions given by the following formulas? (a) \(g(x)=-8 x^{2}+5 x-2\) (b) \(h(x)=-\frac{1}{3} x^{2}+3 x\) (c) \(k(x)=8 x^{2}-5 x-7\) (d) \(j(x)=-\frac{71}{99} x^{2}+210\) (e) \(f(x)=-4 x^{2}-3 x+7\) (f) \(F(x)=-x^{4}+x^{3}+9\)

In this section, it was noted that there is often more than one way to answer a backward question. For example, if the backward question is, "How can we prove that two real numbers are equal?", one possible answer is to prove that their difference equals 0. Another possible answer is to prove that the first is less than or equal to the second and that the second is less than or equal to the first. (a) Give at least one more answer to the backward question, "How can we prove that two real numbers are equal?" (b) List as many answers as you can for the backward question, "How can we prove that a real number is equal to zero?" (c) List as many answers as you can for the backward question, "How can we prove that two lines are parallel?" (d) List as many answers as you can for the backward question, "How can we prove that a triangle is isosceles?"

Following is a statement of a theorem about certain cubic equations. For this theorem, \(b\) represents a real number. Theorem \(\mathbf{A}\). If \(f\) is a cubic function of the form \(f(x)=x^{3}-x+b\) and \(b>1,\) then the function \(f\) has exactly one \(x\) -intercept. Following is another theorem about \(x\) -intercepts of functions: Theorem \(\mathbf{B}\). If \(f\) and \(g\) are functions with \(g(x)=k \cdot f(x),\) where \(k\) is a nonzero real number, then \(f\) and \(g\) have exactly the same \(x\) -intercepts. Using only these two theorems and some simple algebraic manipulations, what can be concluded about the functions given by the following formulas? (a) \(f(x)=x^{3}-x+7\) (b) \(g(x)=x^{3}+x+7\) (c) \(h(x)=-x^{3}+x-5\) (d) \(k(x)=2 x^{3}+2 x+3\) (e) \(r(x)=x^{4}-x+11\) (f) \(F(x)=2 x^{3}-2 x+7\)
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