Understanding the Cardinality of Specific Sets
In this article, we will explore the concept of cardinality and understand the cardinality of two specific sets:1. The set of all real numbers (x) such that (x^2 2).
2. The set of all integers (x) such that (x^2 2).
Set 1: Real Numbers
Let's first consider the set {x | x is a real number and (x^2 2)}. This set can be described as all real numbers (x) that satisfy the equation (x^2 2). Solving the equation (x^2 2), we find two real solutions: (x sqrt{2}) and (x -sqrt{2}). Thus, the set can be expressed as ({sqrt{2}, -sqrt{2}}). The cardinality of this set is 2, as it contains exactly two elements. Formally, this set can also be written as ({x in mathbb{R} | x^2 2}). This set has two elements, which we have shown as ({sqrt{2}, -sqrt{2}}).Set 2: Integers
Next, let's look at the set {x | x is an integer and (x^2 2)}. This set consists of all integers (x) that satisfy the equation (x^2 2). However, there is no integer that, when squared, equals 2. The square root of 2 is approximately 1.414, which is not an integer. Therefore, this set is empty and can be written as (emptyset). The cardinality of this set is 0, as it contains no elements. Formally, this set can be written as ({x in mathbb{Z} | x^2 2}), which is the empty set (emptyset).Cardinality of the Sets
In summary, the cardinality of the two sets is as follows: Cardinality of the set of real numbers (x) such that (x^2 2): 2 Cardinality of the set of integers (x) such that (x^2 2): 0What is Cardinality?
Cardinality in mathematics refers to the number of elements in a set. It is a fundamental concept in set theory and has applications in various areas of mathematics, including computing and data analysis.For the set of real numbers (x) where (x^2 2), the solutions are (x sqrt{2}) and (x -sqrt{2}). Thus, the set contains two elements, and its cardinality is 2.
For the set of integers (x) where (x^2 2), since there are no integer solutions, the set is empty, and its cardinality is 0.