/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 72 Let \(X=\\{1,2\\}, Y=\\{a\\},\) ... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 1: Problem 72

Let \(X=\\{1,2\\}, Y=\\{a\\},\) and \(Z=\\{\alpha, \beta\\} .\) List the elements of each set. $$ X \times Y \times Z $$

Short Answer

Expert verified
The elements of the set \(X \times Y \times Z\) are {(1, a, \alpha), (1, a, \beta), (2, a, \alpha), (2, a, \beta)}.

Step by step solution

01

Understanding the Cartesian Product

The Cartesian product of sets X, Y, Z is derived by pairing each element of X with all elements of Y and Z in an ordered triple. This is represented as \(X \times Y \times Z = \{(x,y,z) | x \in X, y \in Y, z \in Z \}\).
02

Enumerating the Set X

The set X contains the elements {1, 2}. So, we will pair these two elements with the other sets.
03

Enumerating the Set Y

The set Y contains only one element - {a}. We'll pair this one element from Y with each element in set X and set Z.
04

Enumerating the Set Z

Set Z contains two elements - \{\alpha, \beta\}. Each element in X and Y will be paired with each element in Z to provide the overall Cartesian product.
05

Formulating the Cartesian Product

By pairing each element in X with every element in Y and Z, we can formulate the elements of the Cartesian product \(X \times Y \times Z\). These would be: {(1, a, \alpha), (1, a, \beta)}, pairing the first element of X with Y and Z; {(2, a, \alpha), (2, a, \beta))}, pairing the second element of X with Y and Z.

Unlock Step-by-Step Solutions & Ace Your Exams!

  • Full Textbook Solutions

    Get detailed explanations and key concepts

  • Unlimited Al creation

    Al flashcards, explanations, exams and more...

  • Ads-free access

    To over 500 millions flashcards

  • Money-back guarantee

    We refund you if you fail your exam.

Over 30 million students worldwide already upgrade their learning with 91影视!

One App. One Place for Learning.

All the tools & learning materials you need for study success - in one app.

Get started for free

Most popular questions from this chapter

For each pair of propositions \(P\) and \(Q\) . State whether or not \(P \equiv Q\). $$ P=(p \rightarrow q) \rightarrow r, Q=p \rightarrow(q \rightarrow r) $$

Consider the headline: Every school may not be right for every child. What is the literal meaning? What is the intended meaning? Clarify the headline by rephrasing it and writing it symbolically.

Assume that \(\exists x \forall y P(x, y)\) is false and that the domain of discourse is nonempty. Which of must also be false? Prove your answer. $$ \forall x \forall y P(x, y) $$

Determine the truth value of each statement. The domain of discourse is \(\mathbf{R} \times \mathbf{R}\). Justify your answers. $$ \forall x \exists y\left(x^{2} < y+1\right) $$

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

Recommended explanations on Math Textbooks

Probability and Statistics

Read Explanation

Statistics

Read Explanation

Decision Maths

Read Explanation

Mechanics Maths

Read Explanation

Applied Mathematics

Read Explanation

Pure Maths

Read Explanation
View all explanations

What do you think about this solution?

We value your feedback to improve our textbook solutions.

Study anywhere. Anytime. Across all devices.