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Chapter 6: Problem 7

Der Leser mache sich mit der einfachen aber den Anf盲nger oft befremdenden Tatsache vertraut, daB etwa die Ungleichung \(1 \leqslant 2\) ebenso richtig ist wie \(1<2\) und deshalb unbedenklich hingeschrieben werden kann (sie bedeutet nur, daB 1 nicht gr枚\betaer als 2 ist, und dies trifft doch gewiB zu) Die erste Ungleichung ist lediglich ,weniger scharf", sie enth?lt weniger Information als die zweite (sie informiert uns nicht, daB \(1 \neq 2\) ist).

Short Answer

Expert verified
\(1 \leqslant 2\) and \(1 < 2\) are both true, but \(1 < 2\) is more precise.

Step by step solution

01

Understand the Concept of Inequalities

Inequalities express the relative size or order of two values. In mathematics, the symbol \(\leqslant\) is used to denote that a value is less than or equal to another, whereas the symbol \(<\) is used to denote that a value is strictly less than another.
02

Apply the Concept to Example

We are given the inequalities \(1 \leqslant 2\) and \(1 < 2\). The inequality \(1 \leqslant 2\) means "1 is less than or equal to 2", which is true because 1 is indeed less than 2. The inequality \(1 < 2\) means "1 is strictly less than 2", which is also true.
03

Compare the Information Conveyed

The inequality \(1 \leqslant 2\) is less descriptive than \(1 < 2\) because it also includes the possibility of the two values being equal, i.e., \(1 = 2\), which is not true. Therefore, \(1 \leqslant 2\) communicates that 1 is not greater than 2, but doesn't specify if 1 equals 2 or is just less.
04

Conclude on the Use of Both Inequalities

While both inequalities are true statements, \(1 < 2\) gives a clearer picture by excluding the possibility of equality, thus providing more precise information about the relationship between 1 and 2.

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Key Concepts

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

Mathematical Symbols
Mathematical symbols are like the language of math. They help us convey complex ideas in a concise way.
Let's examine a few important symbols:
  • The 'less than' symbol, represented by \(<\), shows that one value is smaller than another.
  • The 'greater than' symbol, \(>\), indicates that one value exceeds another.
  • The 'less than or equal to' symbol, \(\leqslant\), signifies that a value is either smaller or equal to another.
  • The 'greater than or equal to' symbol, \(\geqslant\), means a value is larger or equal to another.
These symbols form the basis of expressing inequalities, which are essential in comparing values. By understanding these symbols, you can effectively read and write mathematical statements.
Less Than or Equal To
The 'less than or equal to' symbol \(\leqslant\) is quite useful in mathematics, offering flexibility in expressing relationships between numbers.
This symbol states that the first number is either less than or on par with the second one.
In the inequality \(1 \leqslant 2\), it tells us that 1 could be less than 2, or, if the values were different, could even be equal. Thus, it covers two possibilities in one statement.This broad interpretation means that using \(\leqslant\) is a safe bet when you're unsure whether equality holds or if you want to include it. It's common in real-world problems, like defining limits or boundaries, as it ensures no values are accidentally excluded.
Strictly Less Than
The 'strictly less than' symbol \(<\) is more specific compared to \(\leqslant\). It clearly states that one value is less than another, excluding any possibility of equality.
This precision can make it very handy.
Take \(1 < 2\) as an example. This inequality not only communicates that 1 is smaller than 2 but also that they are definitely not equal. Such precise information can be crucial in various mathematical contexts, where understanding the exact relationship is necessary. Using \(<\) helps prevent any ambiguity about the sizes of the values, ensuring clarity in mathematical expressions.

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Most popular questions from this chapter

Zeige, daB die nat眉rlichen Zahlen den Peanoschen Axiomen in Nr 3 gen眉gen, wenn man jedem \(n \in \mathbf{N}\) den Nachfolger \(n^{\prime}:=n+1\) zuordnet

Zeige mit Hilfe des Induktionsptinzips, daB jede nichtleere endliche Menge reeller Zahlen ein kleinstes und ein gr枚Btes Element besitzt.

Sei \(m\) eine feste nat眉rliche Zahl. Dann gibt es f眉r jedes nat眉rliche \(n\) nichtnegative ganze Zahlen \(q\) und \(r\), so daB $$ n=q m+r, \quad 0 \leqslant r

Warum ist der folgende,,Beweis" f眉r die Gleichung \(2=1\) falsch? $$ \begin{aligned} a=b \rightarrow a^{2}=a b \rightarrow a^{2}-b^{2}=a b-b^{2} \rightarrow(a+b)(a-b)=b(a-b) \\ & \rightarrow a+b=b \rightarrow 2 b=b \rightarrow 2=1 \end{aligned} $$
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