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Chapter 6: Problem 42

Describe each set in words. a. \(T^{c} \cap C^{c}\) b. \((T \cup C)^{c}\)

Short Answer

Expert verified
Set A represents the intersection of the complements of sets T and C, and Set B represents the complement of the union of sets T and C. Both sets contain all elements that don't belong to either set T or set C, meaning they represent the same set.

Step by step solution

01

Set A: Intersection of Complements

Let's first find the complements of sets T and C. The complement of set T, denoted as \(T^c\), includes all elements in the universal set that are not in set T. Similarly, the complement of set C, denoted as \(C^c\), includes all elements in the universal set that are not in set C. Now, we need to find the intersection between the complements of sets T and C, which is represented as \(T^c \cap C^c\). This intersection includes all elements that are in both \(T^c\) and \(C^c\), meaning all elements that are neither in set T nor in set C. In other words, set A contains all elements that don't belong to either set T or set C.
02

Set B: Complement of the Union of Sets T and C

We will first find the union of sets T and C. The union is represented as \(T \cup C\), and includes all elements that are either in set T, in set C, or in both sets. Next, we need to find the complement of the union of sets T and C, which is represented as \((T \cup C)^c\). This complement includes all elements that are not in the union of sets T and C, meaning all elements that are neither in set T nor in set C. In other words, set B contains all elements that don't belong to either set T or set C. We notice that both set A and set B have the same description, which means they represent the same set. This set includes all elements that are not part of either set T or set C.

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Key Concepts

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

Universal Set
The universal set is a fundamental concept in set theory. It serves as the "big picture" that includes all possible elements under consideration for a particular discussion.
For example, if we are discussing numbers, the universal set might be all real numbers. If we are focusing on students in a class, then the universal set would be all students enrolled in that class. It varies depending on the context of the problem.
Understanding the universal set is important because it provides the reference point for defining other subsets, such as complements and intersections. Every operation, like union or intersection, is done in reference to this universal set.
Complement of a Set
The complement of a set involves all elements that are not in the set within the universal set. It's like asking, "What isn't included in this particular set?"
For a set T, the complement is denoted as \(T^c\) and includes everything in the universal set except the elements in T.
To visualize this, imagine a circle representing the universal set. The area outside the circle representing T within the universal circle is \(T^c\).
Complements are useful for defining differences in sets and understanding what is excluded from a particular group.
Intersection of Sets
The concept of intersection deals with finding common elements between two sets.
For sets T and C, the intersection is represented as \(T \cap C\), which includes all elements that are in both T and C at the same time.
If T is a set of students who like math, and C is a set of students who like science, \(T \cap C\) would be students who like both math and science.
Intersections help identify overlapping characteristics or shared properties of different sets.
Union of Sets
The union of sets combines all elements that belong to either set.
For sets T and C, the union is represented as \(T \cup C\). It includes elements that are in T, in C, or in both.
Continuing with our previous example, if T is a set of math lovers and C is a set of science lovers, \(T \cup C\) includes all students who like math or science or both.
The union operation is key in broadening the scope to see the full range of membership across different sets.

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Most popular questions from this chapter

State whether the statements are true or false. $$ \text { a. }\\{Q\\}=\varnothing $$

An exam consists of ten true-or-false questions. Assuming that every question is answered, in how many different ways can a student complete the exam? In how many ways can the exam be completed if a student can leave some questions unanswered because, say, a penalty is assessed for each incorrect answer?

A warranty identification number for a certain product consists of a letter of the alphabet followed by a five-digit number. How many possible identification numbers are there if the first digit of the five-digit number must be nonzero?

Let \(A=\\{a, e, i, o, u\\}\) and \(B=\\{b, d, e, o, u) .\) Verify by direct computation that \(n(A \cup B)=n(A)+n(B)-n(A \cap B)\).

How many three-letter code words can be constructed from the first ten letters of the Greek alphabet if no repetitions are allowed?
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