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Chapter 1: Problem 12

Prove \((A \cap B) \backslash B=\emptyset\).

Short Answer

Expert verified
Question: Prove that (A 鈭� B) \ B = 鈭�. Answer: To prove (A 鈭� B) \ B = 鈭�, we need to show that any element in the set (A 鈭� B) \ B doesn't exist in B, which means there are no elements in both (A 鈭� B) and B. By analyzing the definitions of intersection and set difference, we see a contradiction when trying to find an element x that belongs to both (A 鈭� B) and B. As a result, there are no such elements, and therefore (A 鈭� B) \ B = 鈭�.

Step by step solution

01

Definitions

Recall the definition of intersection, \(A \cap B = \{x: x \in A \text{ and } x \in B\}\). Also recall the definition of set difference, \(A \backslash B = \{x: x \in A \text{ and } x \notin B\}\).
02

Prove \((A \cap B) \backslash B=\emptyset\)

To show \((A \cap B) \backslash B=\emptyset\), prove that any element \(x\) in \((A \cap B) \backslash B\) doesn't exist in \(B\). Let \(x\) be an arbitrary element of the set \((A \cap B) \backslash B\). By definition of set difference, \(x \in A \cap B\) and \(x \notin B\). Now, let's look at the property of \(x \in A \cap B\). By definition of intersection, this means that \(x\) belongs to both sets \(A\) and \(B\). However, we already know from the set difference definition that \(x \notin B\). This is a contradiction, as \(x\) cannot be in both \(B\) and not in \(B\) at the same time. Hence, there is no such element \(x\) that belongs to \((A \cap B) \backslash B\). Therefore, \((A \cap B) \backslash B=\emptyset\).

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Key Concepts

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

Intersection of Sets
The idea behind the intersection of two sets is relatively straightforward. Imagine that we have two collections of objects, set A and set B. The intersection of these sets, denoted as \(A \cap B\), comprises all elements that are common to both sets. In mathematical terms, it鈥檚 defined as \(A \cap B = \{x: x \in A \text{ and } x \in B\}\).

So, if we take any element that is in both A and B, that element is part of the intersection. For example, if A is the set of all even numbers and B is the set of all integers that are divisible by 4, then the intersection of A and B would include numbers like 4, 8, 12, and so on, since these are both even and divisible by 4.
Set Difference
Moving on to the concept of set difference, we deal with identifying what is unique to one set when compared to another. The set difference \(A \backslash B\) consists of all the elements that are in set A but not in set B. Formally, it鈥檚 described by \(A \backslash B = \{x: x \in A \text{ and } x otin B\}\).

If we continue with our previous sets as an example, the set difference \(A \backslash B\) would contain all even numbers that are not divisible by 4, such as 2, 6, 10, and so on. These are elements that lie in set A but are excluded from B.
Proof by Contradiction
The technique known as proof by contradiction is a powerful method in mathematics used to establish the truth of a statement. Here's how it works: you assume that the statement you're trying to prove is false, and then show that this assumption leads to a contradiction.

When faced with a contradiction, you can conclude that the original assumption (the statement is false) must itself be incorrect. As a result, the statement is true. This technique is especially useful when dealing with complex or abstract concepts where direct proof is challenging.
Empty Set
The empty set, often symbolized by \(\emptyset\), is the set that contains no elements. It's the unique set with zero members and is considered a subset of every set.

Understanding the empty set is crucial in set theory because it acts as the identity element for the operation of union and plays a defining role in the property of the intersection of sets. For example, if we have any set A, the intersection of A with the empty set is always the empty set, \(A \cap \emptyset = \emptyset\), since there are no elements in the empty set to be common with A.

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Most popular questions from this chapter

Prove \(A \cup B=(A \cap B) \cup(A \backslash B) \cup(B \backslash A)\).

Prove \(A \cap(B \cup C)=(A \cap B) \cup(A \cap C)\).

Prove \(A \subset B\) if and only if \(A \cap B=A\).

Define a function on the real numbers by $$ f(x)=\frac{x+1}{x-1} $$ What are the domain and range of \(f ?\) What is the inverse of \(f ?\) Compute \(f \circ f^{-1}\) and \(f^{-1} \circ f\)

Prove the relation defined on \(\mathbb{R}^{2}\) by \(\left(x_{1}, y_{1}\right) \sim\left(x_{2}, y_{2}\right)\) if \(x_{1}^{2}+y_{1}^{2}=x_{2}^{2}+y_{2}^{2}\) is an equivalence relation.
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