Adriano Chikande

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Prove or disprove: $X \times \varnothing=\mathbb{D}$ for every set $X$.

   Prove or disprove: $X \times \varnothing=\mathbb{D}$ for every set $X$.

Discrete Mathematics

Discrete Mathematics

Richard Johnsonbaugh 8th Edition

Chapter 2, Problem 20

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Solved by Expert Adriano Chikande

10/25/2022

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Prove or disprove: $X \times \varnothing=\mathbb{D}$ for every set $X$.

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Key Concepts

Disproof by Counterexample

Disproof by counterexample is a method used to refute universal statements by finding just one instance where the statement fails. In context, since the Cartesian product of any set and the empty set always results in the empty set, showing an instance where this does not match another arbitrary set (such as the set denoted by D) is sufficient to disprove a universal claim that they are equal for every set.

Cartesian Product

The Cartesian product of two sets is defined as the set of all ordered pairs where the first component is an element of the first set and the second is an element of the second set. This fundamental concept in set theory is used to construct product spaces, relations, and functions. It is crucial to understand that if either component set is empty, then there can be no ordered pairs formed, leading to an empty Cartesian product.

Empty Set

The empty set is the unique set that contains no elements. It plays a vital role in set theory because operations involving the empty set often result in predictable outcomes. For example, the Cartesian product of any set with the empty set is always empty, since there is no element in the empty set to pair with elements from the other set.

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