Problem 24 - Discrete Mathematics with Applications

Question

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

Step-by-Step Solution

Answer

The given statement \(\{x\} ∈ \{\{x\}, y\}\) is true, as the set containing only the element x is indeed a member of the set containing the set {x} and the element y.

Step 1: Understanding Set Membership

In set theory, an element is said to be a member of a set if it is included within that set. We use the symbol "∈" to represent membership, e.g. "x ∈ A" means that "x is a member of set A."

Step 2: Analyzing the Given Statement

Now let's analyze the given statement: \(\{x\} ∈ \{\{x\}, y\}\) Here, we are determining if the set containing only the element x ({x}) is a member of the set containing two elements: the set {x} and the element y ({x}, y).

Step 3: Determining Membership

To determine if the given statement is true or false, we must check if the set containing only the element x is indeed a member of the set containing two elements: the set {x} and the element y. In this case, we can clearly see that the set \(\{x\}\) is a member of the set \(\{\{x\}, y\}\) because it is one of the elements in the set.

Step 4: Conclusion

Since the set containing only the element x is indeed a member of the set containing the set {x} and the element y, the given statement: \(\{x\} ∈ \{\{x\}, y\}\), is true.

Key Concepts

set membership

Set membership is a fundamental concept in set theory. It is the relationship between an element and a set, where the element is part of the collection defined by that set. The notation used for expressing set membership is "\(\in\)". For example, if an element "a" is part of a set "A", then this relationship is represented as "\(a \in A\)".
This means "a" is a member of or belongs to "A".
Generally, to verify set membership, one must check if the element in question is present in the collection of elements that make up the set. If it is, then that element is considered a member.

• If an element is part of the set, the statement "element \(\in\) set" holds true.
• If it is not, then the statement is false.

Understanding this simple yet powerful relation is key to solving problems in set theory efficiently.

element of a set

An element is what constitutes a set. Put simply, it is one of the items or members contained in a set. A set can consist of any number of elements and these elements can be anything such as numbers, symbols, or even other sets.
For instance, in the set "\(\{a, b, c\}\)", "a", "b", and "c" are elements of the set.

• Each element is distinct and contributes to the identity of the set.
• It's important to differentiate between an element and a set that contains elements.

In terms of our exercise, the set "\(\{x\}\)" is itself an element of the larger set "\(\{\{x\}, y\}\)".
This distinction helps when dealing with complex set problems, especially those involving nested sets or sets within sets.

membership representation

Membership representation is the way we express the inclusion of an element in a set. We use the symbol "\(\in\)" to denote this relationship in mathematical texts.
This representation is crucial when analyzing statements in set theory, as it provides clarity and precision in understanding what is being asserted.
Let's revisit the exercise statement, "\(\{x\} \in \{\{x\}, y\}\)".

• Here, "\(\{x\}\)" represents an entire set being considered as an element.
• The statement asserts that "\(\{x\}\)" is one of the members of the set "\(\{\{x\}, y\}\)".

Using "\(\in\)" makes it evident that the set "\(\{x\}\)" is indeed part of the set "\(\{\{x\}, y\}\)", proving the statement true.
Clear representation and understanding of membership help students break down and analyze set-based problems competently.