Problem 23 - Discrete Mathematics with Applications

Question

Mark each as true or false. $$\\{x, y\\}=\\{y, x\\}$$

Step-by-Step Solution

Answer

The statement \(\\{x, y\\} = \\{y, x\\}\) is true, as both sets have the same elements, x and y, which satisfies the definition of set equality.

Step 1: Recall the definition of set equality

According to the definition of set equality, two sets A and B are equal, denoted as \(A = B\), if and only if every element in A is also in B, and every element in B is also in A. In other words, both sets have exactly the same elements.

Step 2: Examine the given sets

In this case, the sets are: \(A = \\{x, y\\}\) and \(B = \\{y, x\\}\) We can see that all elements in set A are present in set B, and likewise, all elements in set B are present in set A.

Step 3: Determine the equality of the sets

As we found that the elements of both sets A and B are the same, we can conclude that the statement \(\\{x, y\\} = \\{y, x\\}\) is true.

Key Concepts

Definition of Set Equality

Understanding the concept of set equality is crucial for students delving into the world of mathematics, particularly in set theory. When we talk about set equality, we are referring to the idea that two sets are considered equal if they contain precisely the same elements, no more and no less. The formal definition is succinct: two sets, let's call them Set A and Set B, are equal (denoted as A = B) if every element in Set A is also in Set B and every element in Set B is also in Set A. In simpler terms, it's like saying two bags of fruit are the same if they both contain the exact same kinds of fruit in identical quantities, regardless of the order they're arranged in.

In the exercise example, the statement \(\{x, y\} = \{y, x\}\) is questioned. Following the definition, as both sets contain the same elements, x and y, regardless of their order, we can comfortably declare them equal. It's important to remember that in set theory, the order of elements is immaterial; what counts is purely the presence or absence of an element in the set.

Elements of Sets

Now, what exactly do we mean by an 'element of a set'? In mathematical terms, an element can be anything: a number, a letter (representing numbers or other objects), a symbol, or even another set. Think of a set as a collection or a list, and elements are the items on that list. For example, in the given exercise, x and y are elements of the sets in question.

To tie it back to the exercise, correctly identifying the elements in each set allows us to apply the definition of set equality accurately. Elements must be one and the same for both sets in question for us to deem them equivalent. However, bear in mind that the representation of a set doesn't affect its composition. Whether we write \(\{x, y\}\) or \(\{y, x\}\), the elements remain consistent – the set is unchanged.

Properties of Sets

Sets come with intrinsic properties that help determine relationships such as equality. One of these properties is the aforementioned indifference to element order. To illustrate this with the problem at hand, regardless of whether ‘x’ comes before ‘y’ or after, the sets \(\{x, y\}\) and \(\{y, x\}\) are equal. Another critical property to understand is that sets do not have repeated elements; each element is unique. If you encounter a set written as \(\{a, a, b\}\), this set is effectively the same as \(\{a, b\}\).

By harnessing these properties, it becomes clear that in assessing whether two sets are equal, we merely need to check the elements’ presence and uniqueness. The set's presentation can vary, but its fundamental nature does not. Hence, verifying equality involves a comparison that pays no heed to order or duplicity but focuses entirely on whether the elements within one can be found in the other, and vice versa.