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Chapter 1: Problem 24

If \(A\) and \(B\) are two sets then \(B-(B-A)=\ldots \ldots \ldots\) (a) \((\mathrm{A}-\mathrm{B})-\mathrm{B}\) (b) \(A-(A-B)\) (c) \(\mathrm{A}\) (d) \(\mathrm{B}\)

Short Answer

Expert verified
The short answer is: \[\boxed{A-(A-B)}.\]

Step by step solution

01

Recall the definition of set difference

A set difference is defined as: for two sets A and B, the difference \(A - B\) consists of all the elements of A that are not in B. Using this definition, let's examine each of the given options:
02

Examine option (a)

Option (a) gives the following set operation: \((A-B)-B\). However, this is not equivalent to the given set operation \(B-(B-A)\). So, option (a) is incorrect.
03

Examine option (b)

Option (b) gives the following set operation: \(A-(A-B)\). This looks similar to the given set operation \(B-(B-A)\). Let's check if they are equal. For any element x in the set A: 1. If x is in B, then x is not in A-B. Therefore, x is in A-(A-B). 2. If x is not in B, then x is in A-B. Therefore, x is not in A-(A-B). So, the set \(A-(A-B)\) consists of all the elements of A that are also in B. This is the same as the intersection of A and B or \(A \cap B\). Therefore, option (b) is equivalent to the given set operation \(B-(B-A)\).
04

Examine option (c)

Option (c) gives the set A. This is not equivalent to the given set operation \(B-(B-A)\). So, option (c) is incorrect.
05

Examine option (d)

Option (d) gives the set B. This is not equivalent to the given set operation \(B-(B-A)\). So, option (d) is incorrect. Since option (b) is the only option that is equivalent to the given set operation \(B-(B-A)\), the correct answer is \[\boxed{A-(A-B)}.\]

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