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Chapter 9: Problem 5

Find the intersection of the sets. $$\\{a, b, c, d\\} \cap \va\(\varnothing\)rnothing$$

Short Answer

Expert verified
The intersection of the sets \{a, b, c, d\} and \varnothing is \varnothing.

Step by step solution

01

Identify the sets

Identify the sets to be considered for the intersection operation. Here, the sets are \{a, b, c, d\} and \varnothing.
02

Understand the properties of set intersection

The intersection of any set with the empty set is always the empty set. This is because the intersection includes only those elements that are present in both sets. And since there are no elements in an empty set, the intersection is also an empty set.
03

Apply the intersection operation

Applying the concept from Step 2, it is clear that the intersection of \{a, b, c,d \} and \varnothing is \varnothing, as there are no common elements between these sets.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Intersection of sets
When we talk about the intersection of sets in set theory, we are referring to identifying common elements shared by both sets. Consider it like finding the overlap between two circles in a Venn diagram.

Here are a few key points to understand about intersections:
  • The intersection of two sets is denoted by the symbol \( \cap \).
  • The result of an intersection is a set containing all the elements present in both sets.
  • If no elements are shared, the result is the empty set.
Let's take an example:
If we have two sets \( A = \{a, b, c\} \) and \( B = \{b, c, d\} \), the intersection of \( A \) and \( B \) is \( A \cap B = \{b, c\} \). This is because the elements \( b \) and \( c \) are common in both sets.
Empty set
The empty set, often denoted by \( \varnothing \), is a unique set in mathematics. It contains no elements at all. Think of it as an empty box 鈥� there's nothing inside! This concept is crucial to understand, particularly in operations involving sets.

Consider these characteristics of the empty set:
  • An empty set is considered a subset of all sets, including itself.
  • When intersected with any other set, the result is always an empty set.
  • It plays a vital role in defining certain properties and outcomes in set operations.
For example, if you have a set \( C = \{1, 2, 3\} \) and you find the intersection with \( \varnothing \), the intersection is \( C \cap \varnothing = \varnothing \), because there are no elements in common with the empty set.
Element properties
Elements in a set have specific properties that help define their relationships and operations within set theory. Understanding these properties can clarify operations like intersections and how they relate to sets like the empty set.

Here are some important properties of elements in sets:
  • An element belongs to a set if it is listed as a member of that set.
  • Elements are unique - no duplicates are found within a set.
  • The order of elements does not matter in set theory.
For example, in the set \( D = \{x, y, z\} \), each element \( x, y, z \) is unique and distinguishable. If \( \{x, y, z\} \cap \varnothing \), the result is \( \varnothing \), affirming that no common elements exist between any set and the empty set. Recognizing these properties emphasizes the power and simplicity of using set theory to solve problems.

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Most popular questions from this chapter

The percentage, \(p,\) of defective products manufactured by a company is given by \(|p-0.3 \%| \leq 0.2 \% .\) If \(100,000\) products are manufactured and the company offers a \(\$ 5\) refund for each defective product, describe the company's cost for refunds.

Solve each inequality using a graphing utility. Graph each side separately in the same viewing rectangle. The solution set consists of all values of \(x\) for which the graph of the left side lies above the graph of the right side. $$|2 x-1|>7$$

Describe what is meant by the union of two sets. Give an example.

Without graphing, in Exercises \(55-58,\) determine if each system has no solution or infinitely many solutions. $$\left\\{\begin{array}{l} 3 x+y<9 \\ 3 x+y>9 \end{array}\right.$$

a. Solve: \(2 x-6 \geq-4\) b. Solve: \(5 x+2 \geq 17\) c. Give an example of a number that satisfies the inequality in part (a) and the inequality in part (b). d. Give an example of a number that satisfies the inequality in part (a), but not the inequality in part (b).
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