Problem 9 - Algebra for College Students

Question

Let \(A=\\{1,2,3,4,5,6\\}, B=\\{1,3,5\\}, C=\\{1,6\\},\) and \(D=\\{4\\} .\) Find each set. $$ B \cap \varnothing $$

Step-by-Step Solution

Answer

\(B \cap \varnothing = \varnothing\).

Step 1: Understand the Problem

The problem requires finding the intersection of set B with the empty set \(\varnothing\). The intersection of two sets is the set that contains all elements that are in both sets.

Step 2: Define the Sets

Set \(B\) is given as \[B = \{1,3,5\}\]. The empty set \(\varnothing\) has no elements.

Step 3: Apply the Intersection Operation

The intersection of a set with the empty set \(\varnothing\) is always the empty set. This is because there are no elements that can be common between a set with elements and a set with no elements.

Step 4: Write the Solution

Therefore, \(B \cap \varnothing = \varnothing\).

Key Concepts

set theory

Set theory is a branch of mathematical logic that studies sets, which are collections of objects. Understanding sets is crucial because they form the foundation of many other areas of mathematics.

In set theory, objects in a set are known as elements. Sets are usually denoted with curly braces, like this: \{a, b, c\}. The elements within a set can be anything: numbers, letters, or even other sets!

Some important operations in set theory include:

• Union: Combines all elements from two sets. Denoted as \( A \cup B \).
• Intersection: Finds common elements between two sets. Denoted as \( A \cap B \).
• Difference: Elements in one set but not the other. Denoted as \( A - B \).
• Complement: All elements not in the set, typically within a universal set. Denoted as \( A' \) or \( U - A \).

In our example, we are primarily concerned with the intersection operation.

empty set

The empty set, also known as the null set, is a unique set in set theory because it contains no elements at all. The empty set is denoted by \( \varnothing \) or sometimes by \( \{ \} \).

One important property of the empty set is that its intersection with any other set is always the empty set. This property is crucial to understanding the given exercise.

For example, if \( A = \{1, 2, 3\} \), then \( A \cap \varnothing = \varnothing \). This is because there are no common elements between set \( A \) and the empty set, as the empty set has no elements to contribute.

As seen in the step-by-step solution, we applied this property to solve the exercise involving sets \( B \) and the empty set.

intersection of sets

The intersection of sets is a fundamental operation in set theory. It results in a new set containing all elements that are common to both original sets. The intersection is denoted by the symbol \( \cap \) and can be expressed as:

\[ A \cap B = \{ x \ | \ x \in A \text{ and } x \in B \} \]

For instance, if \( A = \{1, 2, 3\} \) and \( B = \{3, 4, 5\} \), the intersection \( A \cap B = \{3\} \). Here, the only common element between sets \( A \) and \( B \) is 3.

In our exercise, we were asked to find the intersection of set \( B = \{1, 3, 5\} \) with the empty set \( \varnothing \). By now, you should understand that the empty set contains no elements. Hence, there can be no common elements between \( B \) and \( \varnothing \).

Therefore, the intersection \( B \cap \varnothing \) is \( \varnothing \), as there are no elements to be found in both sets.