Problem 34 - Discrete Mathematics with Applications

Question

Rewrite each in words, where UD = set of integers. $$(\forall x)(\forall y)(x+y=y+x)$$

Step-by-Step Solution

Answer

For every integer x and for every integer y, the sum of x and y is equal to the sum of y and x.

Step 1: Identify the symbols

The given expression is \((\forall x)(\forall y)(x+y=y+x)\). Here, the symbol \(\forall\) denotes "for all" or "for every", while x and y are variables representing integers as the UD is the set of integers.

Step 2: Understand the meaning of the expression

The expression \((\forall x)(\forall y)(x+y=y+x)\) can be understood as a claim that for every integer x and for every integer y, the sum of x and y is equal to the sum of y and x.

Step 3: Rewrite in words

To rewrite this expression in words, we'll simply replace the symbols with their meaning in words. The expression can be translated as: "For every integer x and for every integer y, the sum of x and y is equal to the sum of y and x."

Key Concepts

Universal Quantifier

The universal quantifier is a fundamental concept in logic and mathematics, symbolized by \(\forall\), which is read as "for all" or "for every." It is used to express that a statement or property is true for every member of a specific set.
For instance, when we write \(\forall x\), we are saying that the subsequent statement applies to all possible values of \(x\) within a given set. In the context of our original exercise, the set is the set of integers, meaning the statement has to hold true for every integer.
This allows us to make comprehensive, sweeping statements in mathematics that would otherwise be cumbersome to articulate. It's a tool for generalization and is crucial in proving theorems and other mathematical principles as it succinctly captures the notion of universal truth across a domain.

Integers

Integers are a set of numbers that include all whole numbers, both positive and negative, as well as zero. This set is often represented by the letter \(\mathbb{Z}\). Specifically, integers include numbers like \(-3, -2, -1, 0, 1, 2, 3,\) and so on, extending infinitely in both the positive and negative directions.
Integers are fundamental in mathematics because they form the building blocks of number systems. They appear naturally when counting objects and measuring quantities where fractions and decimals are not necessary.
Given their straightforward nature, integers play a critical role in many branches of mathematics, including algebra and number theory. In our original exercise, the universal quantifier applies to these integers, meaning the statement involving addition must be true no matter which integers are chosen.

Mathematical Expressions

Mathematical expressions are mathematical phrases that can contain numbers, variables, and operations. They are the language through which math communicates relationships and operations.
In the original exercise, the expression \(x + y = y + x\) is a simple example of a mathematical expression where addition is the operation. This particular expression is an illustration of the commutative property, which states that the order of addition does not affect the result.
Understanding mathematical expressions is key to solving equations and understanding the wide array of mathematical properties. They often serve as the foundation of the more complex structures in mathematics, such as equations and inequalities. Expressions are essential in defining the relationships that provide solutions to countless mathematical problems.