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Chapter 5: Problem 8

Show that the set \(\\{1,2,4,8,16, \ldots\\}\) of powers of 2 is representable.

Short Answer

Expert verified
The set is representable because each element is a power of 2 expressible as \(2^n\) for non-negative integers \(n\).

Step by step solution

01

Understanding the Set

The set given is \(\{1,2,4,8,16,\ldots\}\). Each element in this set is a power of 2, which can be expressed generally as \(2^n\), where \(n\) is a non-negative integer.
02

Defining Representable Sets

In mathematical terms, a set is representable if it can be described or generated by a certain rule or formula. In this case, a set of numbers is considered representable if there's a systematic way to list elements, and each element can be expressed in a consistent formula.
03

Formulating the General Expression

Let's describe the set \(\{1,2,4,8,16,\ldots\}\) using an explicit formula. Each element is presented as \(2^n\), where \(n = 0, 1, 2, 3, 4, \ldots\). Thus, we can write that any element in the set is of the form \(a_n = 2^n\).
04

Proving Systematic Representation

The expression \(a_n = 2^n\) demonstrates that each element can be systematically generated based on the non-negative integer \(n\). Therefore, since each element matches this rule, the set is representable by this consistent mathematical expression.

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Key Concepts

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

Powers Of Two
The set \(\{1, 2, 4, 8, 16, \ldots\}\) is made up of numbers that are powers of two. In mathematical terms, this means each element can be expressed as \(2^n\), where \(n\) represents a non-negative integer (0, 1, 2, 3, and so on). To break it down:
  • \(2^0 = 1\)
  • \(2^1 = 2\)
  • \(2^2 = 4\)
  • \(2^3 = 8\)
  • \(2^4 = 16\)
...and it continues infinitely. These powers of two are crucial in many areas like computer science, particularly because digital systems use base-2 numeral systems. Each power of two represents a fundamental building block in this context. Whenever we talk about powers of two, the pattern is not only predictable but also goes on indefinitely as we can always find a subsequent power by simply multiplying the current value by 2.
Representable Sets
A set is considered representable if there is a specific rule or formula used to create it. For the set \(\{1, 2, 4, 8, 16, \ldots\}\), it is representable because we can generate every element using the consistent formula \(2^n\). The concept of representability means:
  • We can predict and list elements through a defined mathematical expression.
  • There is an inherent orderliness to how each element is determined.
In essence, representable sets have a clear structure. We know exactly what to do to find the next element in the set. This shows us a systematic way to identify and understand members of the set at any level of complexity, as long as they conform to the established rule. Representability ensures that we aren't just stumbling upon numbers randomly, but rather we are strategically and predictively generating them.
Systematic Representation
Systematic representation is about having an organized method to express every element in a set. For our powers of two set, this means using the formula \(a_n = 2^n\). This systematic approach is beneficial because:
  • It provides a clear formulaic representation for any element within the set.
  • It simplifies the process of identifying each component by relating it to the non-negative integer \(n\).
  • We systematically build the set by applying the rule \(2^n\) for increasing values of \(n\).
With a systematic representation, we are not guessing whether a number belongs to the set, we know precisely where it fits based on its formulaic position. This method also helps in computations, as it allows computers or algorithms to generate numbers efficiently by following a straightforward numerical blueprint.

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Most popular questions from this chapter

We will say that a formula \(\phi(x)\) with one free variable is positively numeralwise determined if, for each \(a \in \mathbb{N}\), if \(\mathfrak{N} \models \phi(a)\) then \(N \vdash \phi(\bar{a})\). Say \(\phi(x)\) is numeralwise determined if both \(\phi(x)\) and \(\neg \phi(x)\) are positively numeralwise determined. Prove that \(\phi\) represents a set \(A\) if and only if \(\phi\) defines \(A\) and \(\phi\) is numeralwise determined. To reiterate, a set \(A\) is representable if and only if \(A\) has a numeralwise determined definition.

(a) Show that \(A \subseteq \mathbb{N}\) is semi-calculable if and only if \(A\) is listable, where a set is listable if there is a computer program \(L\) such that \(L\) prints out, in some order or another, the elements of \(A\). (b) Show \(A \subseteq \mathbb{N}\) is calculable if and only if \(A\) is listable in increasing order.

We defined calculable functions and semi-calculable sets in this section, but the definitions are not set off in their own block and given fancy numbers, like "Definition 5.4.2." Why didn't we make the definitions official-looking like that?

Suppose that \(A \subseteq \mathbb{N}\) is representable and represented by the formula \(\phi(x)\). Suppose also that \(B \subseteq \mathbb{N}\) is representable and represented by \(\psi(x)\). Show that the following sets are also representable, and find a formula that represents each: (a) \(A \cup B\) (b) \(A \cap B\) (c) The complement of \(A,\\{x \in \mathbb{N} \mid x \notin A\\}\)

Write a \(\Delta\) -definition for the set \(D I V I D E S .\) So you must come up with a formula with two free variables, Divides \((x, y)\), which has the property that \(\mathfrak{N} \models\) Divides \((\bar{a}, \bar{b})\) if and only if \(a\) is a factor of \(b\).
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