Question

If \(\mathrm{a}=\\{1,2,3,4,5\\}\) and \(\mathrm{b}=\\{2,3,4,5,6\\}\), find \(\mathrm{a} \cup \mathrm{b}\).

Short answer

The union of the given sets a = {1, 2, 3, 4, 5} and b = {2, 3, 4, 5, 6} is a ∪ b = {1, 2, 3, 4, 5, 6}.

Step-by-step solution

Identifying the given sets

The given sets are: a = {1, 2, 3, 4, 5} and b = {2, 3, 4, 5, 6}

Finding the union of the sets

To find the union a ∪ b, we will combine all the elements of set a and set b without any repetition of elements. Let's look at the elements in both sets: - Set a has the elements: 1, 2, 3, 4, 5 - Set b has the elements: 2, 3, 4, 5, 6 Notice that the elements 2, 3, 4, and 5 are present in both sets.

Combining the elements without repetition

Now, we will combine all the elements from set a and set b without repeating any element. Here is the union of the two sets without any repetition: a ∪ b = {1, 2, 3, 4, 5, 6} Therefore, the union of set a and set b is {1, 2, 3, 4, 5, 6}.

Key concepts

Union of Sets

The concept of the union of sets is integral to understanding set theory. It involves combining all elements that exist in either of the two sets being considered. In other words, if you have two sets, say set \( a \) and set \( b \), their union, denoted \( a \cup b \), includes every unique element found in either set or both. It is important to note that when forming this union, the repetition of common elements between sets is not allowed.

To perform a union of sets, follow these steps:

• List all elements of the first set.
• List all elements of the second set.
• Combine the lists, but ensure no duplicates remain.

The result is a new set containing all distinct elements from both sets. For instance, if \( a = \{1, 2, 3, 4, 5\} \) and \( b = \{2, 3, 4, 5, 6\} \), the union \( a \cup b \) would yield \( \{1, 2, 3, 4, 5, 6\} \).

Elements of a Set

Understanding elements of a set is foundational in set theory. An element is any individual object contained within a set. Sets are generally denoted by listing their elements between curly braces \(\{\}\). Each element within a set is unique, meaning sets do not take into account how often an item appears.

When referring to specific elements of a set, note the distinction from the overall set as a complete collection. For example, in the set \( a = \{1, 2, 3, 4, 5\} \), the numbers 1, 2, 3, 4, and 5 are the elements. They represent what constitutes the set, demonstrating that the set can be conceived as a collection of distinct elements.

It's essential to recognize that when discussing elements, what matters is whether or not an element is present in the set, not how many times it is listed.

Repetition in Sets

Repetition in sets is a topic that clarifies why certain rules exist in set theory. By definition, a set contains only distinct elements. This intrinsic property ensures that there is never a need to list an element more than once within the same set. Even if an element appears multiple times during the formation process, it will only be included once in the final set.

For example, if you are forming a union of two sets, all common elements are written only once. In the example where \( a = \{1, 2, 3, 4, 5\} \) and \( b = \{2, 3, 4, 5, 6\} \), elements 2, 3, 4, and 5 appear in both sets, but in the union \( a \cup b \), they are written just once, creating \( \{1, 2, 3, 4, 5, 6\} \).

This approach simplifies operations in set theory, ensuring clarity and consistency throughout various mathematical processes involving sets.