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Chapter 1: Problem 93

List the members of \(\mathcal{P}(\\{a, b\\})\). Which are proper subsets of \(\\{a, b\\} ?\)

Short Answer

Expert verified
The power set \(\mathcal{P}(\{a, b\})\) includes \(\{\}, \{a\}, \{b\}, \{a, b\}\). The proper subsets of \(\{a, b\}\) are \(\{\}, \{a\}, \{b\}\).

Step by step solution

01

Identify the elements in the set

The given set here is \(\{a, b\}\). So, the elements are 'a' and 'b'.
02

List members of the power set

The power set of a set with n elements has \(2^n\) elements. Here, n=2. So there will be \(2^2 = 4\) elements in the power set \(\mathcal{P}(\{a, b\})\). The members are: \(\{\}, \{a\}, \{b\}, \{a, b\}\). We start by the empty set, then every element individually, and lastly the original set itself.
03

Identify Proper Subsets

Proper subsets are all the subsets of the set that not equal to the set itself. The proper subsets of \(\{a, b\}\) appears to be: \( \{\},\{a\}, \{b\} \)

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Key Concepts

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

Proper Subset
A proper subset is a concept found in set theory where one set, called a subset, is completely contained within another set, but is not identical to that original set. Let's break it down a bit.
  • A set \( A \) is a proper subset of a set \( B \) if every element of \( A \) is in \( B \), and there is at least one element in \( B \) that is not in \( A \).
  • This means if you take the set \( \{ a, b \} \) from our example, its proper subsets are \( \{ \} \), \( \{ a \} \), and \( \{ b \} \).
  • The subset \( \{ a, b \} \) is not proper because it is equal to itself.
The primary function of identifying proper subsets is to understand the variety and depth of relationships within elements. This idea is crucial when analyzing and organizing information in discrete structures.
Set Theory
Set theory is the fundamental language of mathematics, and it serves as a basis for discrete mathematics. It deals with the collection of objects called "sets" and the relationships between them. These concepts are vital in understanding larger mathematical structures.
  • Every object in a set is called an element or member.
  • For instance, in the set \( \{ a, b \} \), 'a' and 'b' are elements.
  • Sets can be described in different ways, such as listing its members or by defining a common property of its members.
Set theory incorporates various operations such as unions, intersections, and differences that assist in deepening the comprehension of how sets interact. It鈥檚 a playground for mathematical theory with applications across various fields, from computer science to logic and more.
Discrete Mathematics
Discrete mathematics is an area of mathematics that deals with countable, distinct, and separate objects. Unlike continuous mathematics, which deals with smooth objects that require the concept of limits, discrete math focuses on distinct objects.
  • Some key topics in discrete math include set theory, logic, and graph theory.
  • It's particularly useful in computer science, where data is often structured in countable or distinct packets.
  • Understanding concepts such as proper subsets and power sets is crucial as they are foundational tools in this subject.
Through discrete mathematics, complex problems can be broken down into finite, manageable components, making it easier to solve real-world problems efficiently. Thus, these concepts form an indispensable part of creating algorithms and constructing logical proofs.

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Most popular questions from this chapter

Let the universe be the set \(Z^{+} .\) Let \(X=\) \\{1,2,3,4,5\\} and let \(Y\) be the set of positive, even integers. In set builder notation, \(Y=\left\\{2 n \mid n \in Z^{+}\right\\} .\) In Exercises \(18-27,\) give a mathematical notation for the set by listing the elements if the set is finite, by using set-builder notation if the set is infinite, or by using a predefined set such as \(\varnothing\). Describe \(\bar{Y}\) in words.

Let \(P(x, y)\) be the propositional function \(x \geq y .\) The domain of discourse is \(\mathbf{Z}^{+} \times \mathbf{Z}^{+} .\) Tell whether each proposition is true or false. $$ \forall x \exists y P(x, y) $$

Verify the second of De Morgan's laws, \(\neg(p \wedge q) \equiv \neg p \vee \neg q\).

Let the universe be the set \(U=\\{1,2,3, \ldots, 10\\}\) Let \(A=\\{1,4,7,10\\}, B=\\{1,2,3,4,5\\},\) and \(C=\\{2,4,6,8\\} .\) List the elements of each set. $$A \cup B$$

Formulate the symbolic expression in words using. \(p:\) You play football. q: You miss the midterm exam. \(r:\) You pass the course. $$ (p \wedge q) \vee(\neg q \wedge r) $$
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