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Chapter 2: Problem 22

Show that if \(A\) and \(B\) are sets with \(A \subseteq B,\) then a) \(A \cup B=B .\) b) \(A \cap B=A\)

Short Answer

Expert verified
a) A 鈭� B = B. b) A 鈭� B = A.

Step by step solution

01

- Understanding the problem

There are two parts to this exercise. We need to show two things. First, that the union of sets A and B is equal to set B if A is a subset of B. Second, that the intersection of sets A and B is equal to set A if A is a subset of B.
02

- Define the given condition

Given that set A is a subset of set B, denoted by A 鈯� B, means that every element of A is also an element of B.
03

- Prove part (a): A 鈭� B = B

Start with the union definition: the union of sets A and B, A 鈭� B, includes all elements that are in either A, B, or both. Since A 鈯� B, all elements of A are already in B. Therefore, A 鈭� B includes all elements of B, which means A 鈭� B = B.
04

- Prove part (b): A 鈭� B = A

Start with the intersection definition: the intersection of sets A and B, A 鈭� B, includes all elements that are common to both A and B. Since A 鈯� B, all elements in A are also in B, meaning A 鈭� B will include all elements of A. Therefore, A 鈭� B = A.

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Key Concepts

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

Subset
A subset is a set where all its elements are contained within another set. For example, if we have two sets, A and B, and every element in A is also in B, we say A is a subset of B. We write this as

\( A \subseteq B \).

Understanding subsets is crucial because it sets the foundation for comparing sets. The concept tells us that no element exists in A that isn't in B. It鈥檚 like saying all students in the math club are students at the school, but not all students at the school are in the math club.

  • To remember this, think: if you can list out all members of set A and find each one in set B, then A is a subset of B.
Union of Sets
The union of two sets is a set containing all elements from both sets. Using the union symbol (\( \cup \)), the union of sets A and B, written as \( A \cup B \), includes every element that is in A, or B, or both.

  • This can be visualized as pooling together elements from both sets into one big set.


When we say \( A \subseteq B \), then all elements of A are already in B. So, when we take the union \( A \cup B \), we essentially are just collecting all elements of B, since A does not add any new elements that aren鈥檛 already in B. This is why \( A \cup B = B \) when A is a subset of B.

  • Visualize this with a simple example: If A={1, 2} and B={1, 2, 3}, then \( A \cup B \) simply equals B since A contributes no new elements.
Intersection of Sets
The intersection of two sets is a set containing only the elements that are in both sets. Using the intersection symbol (\( \cap \)), the intersection of sets A and B, written as \( A \cap B \), includes every element that is shared between both sets.

  • Think of this as finding common ground between two sets.


Given that \( A \subseteq B \), all elements in A are automatically in B. So when we take \( A \cap B \), we are effectively listing out all elements of A, since those are the elements common to both. This explains why \( A \cap B = A \) when A is a subset of B.

  • Picture this with an example: If A={1, 2} and B={1, 2, 3}, then \( A \cap B \) equals A since all elements of A are already in B.

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Most popular questions from this chapter

Suppose that the universal set is \(U=\\{1,2,3,4,\) \(5,6,7,8,9,10 \\} .\) Express each of these sets with bit strings where the \(i\) th bit in the string is 1 if \(i\) is in the set and 0 otherwise. a) \(\\{3,4,5\\}\) b) \(\\{1,3,6,10\\}\) c) \(\\{2,3,4,7,8,9\\}\)

Use the Schr枚der-Bernstein theorem to show that \((0,1)\) and \([0,1]\) have the same cardinality.

Let \(a_{n}\) be the \(n\) th term of the sequence \(1,2,2,3,3,3,\) \(4,4,4,4,5,5,5,5,5,6,6,6,6,6,6, \ldots,\) constructed by including the integer \(k\) exactly \(k\) times. Show that \(a_{n}=\left\lfloor\sqrt{2 n}+\frac{1}{2}\right\rfloor\)

Show that if \(A\) and \(B\) are finite sets, then \(A \cup B\) is a finite set.

What is the value of each of these sums of terms of a geometric progression? $$ \begin{array}{ll}{\text { a) } \sum_{j=0}^{8} 3 \cdot 2^{j}} & {\text { b) } \sum_{j=1}^{8} 2^{j}} \\ {\text { c) } \sum_{j=2}^{8}(-3)^{j}} & {\text { d) } \sum_{j=0}^{8} 2 \cdot(-3)^{j}}\end{array} $$
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