Special Sets

Section 5.4 Special Sets

We introduce notation for some sets that we will use throughout this course.

To make the notation unique and recognizable, we denote some special sets using specific capital letters \(\A\text{,}\) \(\N\text{,}\) \(\PP\text{,}\) \(\W\text{,}\) and \(\Z\) in a font called blackboard bold. While the elements of some of the sets in the following definition the importance of others will become clear later.

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Definition 5.23.

We define the following sets:

The set \(\Z:=\{\dots,-3,-2,-1,0,1,2,3\dots\}\) is the set of integers.
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The set \(\W=\{0,1,2,3,\dots\}\) is the set of whole numbers. It is also known as the set of non-negative integers.
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The set \(\N:=\{1,2,3,\dots\}\) is the set of natural numbers.
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The set \(\PP\) is the set of prime numbers.
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For \(n\in\N\) we define \(\Z_n:=\{0,1,2,\dots,n-1\}\text{.}\) We read \(\Z_n\) as “z n.”
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For \(n\in\N\) we define \(\Z_n^\otimes:=\{1,2,\dots,n-1\}\text{.}\) We read \(\Z_n^\otimes\) as “z n without zero.”
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The set \(\A:=\{\cspace,\mathtt{a},\mathtt{b},\mathtt{c},\dots,\mathtt{x},\mathtt{y},\mathtt{z}\}\) is the set of characters
¹
For technical reasons we use the symbol \(\cspace\) instead of the character space to separate words.
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We have already encountered the integers and natural numbers in Part I and now have a handy way of talking about all of these as the set of integers \(\Z\) and the set of natural numbers \(\N\text{.}\) The set \(W\) contains all elements of \(\N\) and also the number \(0\text{.}\)

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We will discuss the prime numbers in Chapter 10.

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We will come across the sets \(\Z_n\) and \(\Z_n^\otimes\) throughout the remainder of this course. The set \(\Z_n\) is of interest to us because it contains exactly the numbers that occur as remainders when dividing by \(n\text{,}\) that is

\begin{equation*} \Z_n = \{ x\fmod n\mid x\in\Z\}. \end{equation*}

In Chapter 14 we will consider operations on the elements of these sets.

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We use the set of characters \(\A\) to encode characters as numbers in Section 8.1 and to encode texts as numbers in Section 12.3. We will also use it when encrypting messages in Section 8.3, Section 8.4, and Section 16.3. To distinguish characters from variables we write characters in a typewriter font.

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(a) The set of integers \(\Z\) on the number line.

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(b) The set of whole numbers \(\W\) on the number line.

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(d) The set of \(\Z_7=\{0,1,2,3,4,5,6\}\) on the number line.

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(e) The set of \(\Z_7^\otimes=\{1,2,3,4,5,6\}\) on the number line.

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Figure 5.24. The sets \(\Z\text{,}\) \(\W\text{,}\) \(\N\text{,}\) \(\Z_7\text{,}\) and \(\Z_7^\otimes\) on the number line line

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In Figure 5.24 we show sets from Definition 5.23 on the number line. For the sets \(\Z_n\) and \(\Z_n^\otimes\) we chose the special case \(n=7\text{.}\) In Example 5.25 we consider the sets \(\Z_n\) and \(\Z_n^\otimes\text{,}\) where \(n = 6\text{.}\)

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Example 5.25. Reading \(\Z_6\) and \(\Z_6^\otimes\).

\(\Z_6=\{0,1,2,3,4,5\}\) is read “Z 6 is equal to the set containing 0,1,2,3,4, and 5.”
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\(\Z_6^\otimes=\{1,2,3,4,5\}\) is read “Z 6 without 0 is equal to the set containing 1,2,3,4, and 5.”
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It is always interesting to investigate what a definition means for the border line cases. As \(\Z_n\) and \(\Z_n^\otimes\) are define for all natural numbers \(n\) we check what the definition means for the two lowest allowed values of \(n\text{,}\) namely \(n=1\) and \(n=2\text{.}\)

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Example 5.26. \(\Z_n\) and \(\Z_n^\otimes\) for \(n\in\{1,2\}\).

We consider \(\Z_n\) and \(\Z_n^\otimes\) and \(n\in\{1,2\}\) (this means we will look at the cases when \(n =1\) and when \(n =2\)).

\(\displaystyle \Z_1=\{0\}\)
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\(\displaystyle \Z_1^\otimes=\{\}\)
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\(\displaystyle \Z_2=\{0,1\}\)
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\(\displaystyle \Z_2^\otimes=\{1\}\)
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Make sure that you have understood the notation for special sets by giving them in roster form.

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Checkpoint 5.27. Write a special set in roster form.

Give the set in roster form.

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\(\mathbb{Z}_{3} = \lbrace\)\(\rbrace\)

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Checkpoint 5.28. Write another special set in roster form.

Give the set in roster form.

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\(\mathbb{Z}_{10}^\otimes = \lbrace\)\(\rbrace\)

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In the video in Figure 5.29 we recall the definitions of special sets and show how statements that we encountered in Chapter 1 can be reformulated in a shorter way using set notation.

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Figure 5.29. Special Sets by Matt Farmer and Stephen Steward

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Subsection Formulating Statements with Sets

We can use set notation and the special sets defined above to give shorter formulations of statements from Section 1.3. Essentially we are replacing “let \(a\) be an integer” by “let \(a\in\Z\)”.

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Example 5.30. All natural numbers are positive in set notation.

For all \(n\in\N\) we have \(n>0\text{.}\) (compare Problem 1.36)

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The commutative property of addition for integers from Example 1.38 becomes:

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Example 5.31. Commutative property in set notation.

For all \(a\in\Z\) and \(b\in\Z\) we have \(a+b = b+a\text{.}\)

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Likewise the distributive property (compare Example 1.39 can be written as:

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Example 5.32. Distributive property in set notation.

For all \(a\in\Z\text{,}\) \(b\in\Z\text{,}\) and \(c\in\Z\) we have \(a\cdot(b+c)=(a\cdot b) + (a\cdot c)\text{.}\) p

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Theorem 1.48 states that for all integers there exists an additive inverse. We can write this theorem as:

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Theorem 5.33.

For all \(a\in\Z\) there is a \(b\in\Z\) such that \(a+b=0\text{.}\)

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Finally we reformulate Theorem 1.59 and Theorem 1.60 and Theorem 1.65 using set notation

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Theorem 5.34. Properties of exponentiation in set notation.

Let \(a\in\Z\) and \(b\in\Z\) and let \(m\in\N\) and \(n\in\N\text{.}\) Then

\(\displaystyle (b^m)\cdot (b^n)=b^{(m+n)}\)
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\(\displaystyle {(b^m)}^n=b^{(m\cdot n)}\)
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\(\displaystyle (a\cdot b)^n=(a^n)\cdot (b^n)\)
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In Checkpoint 5.35 and Checkpoint 5.36 we formulate some statements about integer operations in set notation with special sets. Decide whether they are true or false. Disprove `for all’ statements with a counterexample. When a `there exists’ statement is true, give values of the variables for which it is true.

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Checkpoint 5.35. `For all’ statements with special sets.

Decide whether the following statements are true or false.

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If the statement is false give a counterexample, otherwise leave the box empty.

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(i) For all \(a\in\mathbb{Z}\) and \(b\in\mathbb{Z}\) we have \(a+b=b+a\text{.}\)

select
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The statement is true.
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The statement is false.
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Give a counterexample if the statement is false: \(a=2\text{,}\) \(b=\)

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(ii) For all \(a\in\mathbb{Z}\) and \(b\in\mathbb{Z}\) and \(c\in\mathbb{Z}\) we have \(a-(b-c)=(a-b)-c\text{.}\)

select
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The statement is true.
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The statement is false.
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If the statement is false, give a counterexample: \(a=2\text{,}\) \(b=11\text{,}\) \(c=\)

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(iii) For all \(a\in\mathbb{Z}\) and \(b\in\mathbb{Z}\) and \(c\in\mathbb{Z}\) we have \(a\cdot(b+c)=a\cdot b+a \cdot c\text{.}\)

select
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The statement is true.
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The statement is false.
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If the statement is false, give a counterexample: \(a=\), \(b=11\text{,}\) \(c=2\text{.}\)

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Checkpoint 5.36. `There exists’ statements with special sets.