**Exercise 1.** For any element $ x $ and set $ A $, prove that the sets $ A $ and $ \{x\} \times A $ have the same cardinality.In this exercise, you are asked to demonstrate the concept of cardinality in sets. Specifically, you need to show that the set $ A $ and the Cartesian product of the singleton set $\{x\}$ with $ A $ have an equal number of elements (same cardinality). This involves constructing a bijection between the two sets to establish their cardinality equivalence.

Result: Let A and B be any sets with the cardinality |A|=m and |B|=n, then the cardinality of A×B…

Answered: Exercise 1. For any element x and set…

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Exercise 1. For any element x and set A, prove that the sets A and {x} × A have the same cardinality.

Advanced Engineering Mathematics

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

Section: Chapter Questions

Problem 1RQ

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**Exercise 1.** For any element $ x $ and set $ A $, prove that the sets $ A $ and $ \{x\} \times A $ have the same cardinality.

In this exercise, you are asked to demonstrate the concept of cardinality in sets. Specifically, you need to show that the set $ A $ and the Cartesian product of the singleton set $\{x\}$ with $ A $ have an equal number of elements (same cardinality). This involves constructing a bijection between the two sets to establish their cardinality equivalence.

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