Problem 22 - Discrete Mathematics with Applications

Question

Rewrite each sentence symbolically, where the UD consists of real numbers. There are real numbers \(x\) and \(y\) such that \(x=2 y\)

Step-by-Step Solution

Answer

\(\exists x, y \in \mathbb{R}: x = 2y\)

Step 1: Identify the Variables

In the given sentence, we are working with the real numbers x and y.

Step 2: Determine Appropriate Quantifiers

We need to express that there exist real numbers x and y satisfying the given condition. We use the existential quantifier (∃) to indicate this.

Step 3: Rewrite Sentence Symbolically

Using the existential quantifier, we can rewrite the sentence symbolically as: \[\exists x, y \in \mathbb{R}: x = 2y\]

Key Concepts

Existential Quantifier

In symbolic logic, particularly within mathematics, the existential quantifier plays a crucial role in expressing the idea that there is at least one element in a specific set that satisfies a particular condition. When dealing with real numbers, which make up a continuous and infinite set, the existential quantifier is denoted by the symbol \( \exists \). This symbol helps us formulate statements about the existence of an unknown value within the universe of discourse—in this case, the set of all real numbers, symbolized as \( \mathbb{R} \).

For instance, to convey that there are real numbers \(x\) and \(y\) such that \(x = 2y\), we use the existential quantifier as \(\exists x, y \in \mathbb{R}: x = 2y\). It signifies that there are specific values for \(x\) and \(y\) in the set of real numbers that fulfill the equation given. This concept is foundational to understanding more complex mathematical expressions and theories that involve variable relationships.

Real Numbers

The set of real numbers is one of the most fundamental concepts in mathematics. Real numbers are the combination of rational and irrational numbers, covering every possible number along the continuous number line we can possibly think of. They include the integers, positive and negative fractions, decimals, and numbers like \(\pi\) and \(e\) that cannot be expressed as a simple fraction.

Real numbers are represented symbolically as \( \mathbb{R} \) and play a central role in mathematical expressions, especially when quantifying variables or constants. The symbolic logic exercise discussed involves the set of real numbers as its universe of discourse. This means that when we say \( \exists x, y \in \mathbb{R} \), we imply x and y are drawn from the vast, unending set of real numbers. Students should be comfortable with the concept of real numbers to fully comprehend and manipulate mathematical expressions and quantifiers.

Mathematical Expressions

Mathematical expressions are combinations of numbers, variables, operations, and sometimes constants that represent a particular quantity or a relationship between quantities. Understanding how to create, interpret, and manipulate these expressions is key in mastering many areas of mathematics.

In the context of our initial statement, \(\exists x, y \in \mathbb{R}: x = 2y\) itself is a mathematical expression that conveys a relationship between two real numbers \(x\) and \(y\). It asserts that \(x\) is double the value of \(y\). Such expressions are not only the language of algebra but also serve as the stepping stones to more advanced topics in calculus, statistics, and other fields of mathematics. Simplifying, analyzing, and solving these kinds of expressions enable students to tackle a vast array of mathematical problems.