Let A and B be sets. If A X = B X = and A X = B X for some set X, show that A = B.

Let A and B be sets. If $A \cap X = B \cap X = \phi$ and $A \cup X =…

Question 47 of 47advanced🔍 AnalyzeLong Answer5 marks

Let A and B be sets. If $A \cap X = B \cap X = \phi$ and $A \cup X = B \cup X$ for some set X, show that $A = B$ .

Correct Answer

We are given that A and B are sets such that $A \cap X = B \cap X = \phi$ and $A \cup X = B \cup X$ . We need to show that A = B. We will use the hint provided: $A = A \cap (A \cup X)$ and $B = B \cap (B \cup X)$ .

First, let us prove that A is a subset of B. We start with the expression for A:

$A = A \cap (A \cup X)$

Since $A \cup X = B \cup X$ (given), we can substitute this into the equation:

$A = A \cap (B \cup X)$

Using the distributive law, we distribute the intersection over the union:

$A = (A \cap B) \cup (A \cap X)$

We are given that $A \cap X = \phi$ . Substituting this value:

$A = (A \cap B) \cup \phi$

$A = A \cap B$

Since $A = A \cap B$ , it implies that A is a subset of B ( $A \subset B$ ).

Next, let us prove that B is a subset of A. We start with the expression for B:

$B = B \cap (B \cup X)$

Since $B \cup X = A \cup X$ (given), we substitute this into the equation:

$B = B \cap (A \cup X)$

Using the distributive law:

$B = (B \cap A) \cup (B \cap X)$

We are given that $B \cap X = \phi$ . Substituting this value:

$B = (B \cap A) \cup \phi$

$B = B \cap A$

Since $B = B \cap A$ , it implies that B is a subset of A ( $B \subset A$ ).

Since we have shown that $A \subset B$ and $B \subset A$ , we can conclude that the sets are equal. Therefore, A = B is proven.

Exercise: Miscellaneous Exercise on Chapter 1 | Q: 9 | (Chapter: 22)

For More Understanding

Explanation

The solution follows the hint provided in the textbook context, utilizing the absorption property $A = A \cap (A \cup X)$ and the distributive law. By substituting the given conditions ( $A \cap X = \phi$ and $A \cup X = B \cup X$ ), the proof demonstrates that A is a subset of B and B is a subset of A, which logically leads to the conclusion that A = B.

Solution Steps

Step 1: Start with the relation $A = A \cap (A \cup X)$ as provided in the hint.
Step 2: Substitute $A \cup X$ with $B \cup X$ (since they are given as equal) to get $A = A \cap (B \cup X)$ .
Step 3: Apply the distributive law to expand: $A = (A \cap B) \cup (A \cap X)$ .
Step 4: Substitute $A \cap X = \phi$ (given) to get $A = (A \cap B) \cup \phi$ , which simplifies to $A = A \cap B$ , proving $A \subset B$ .
Step 5: Repeat the process for B: Start with $B = B \cap (B \cup X)$ , substitute $B \cup X = A \cup X$ , and apply the distributive law to get $B = (B \cap A) \cup (B \cap X)$ .
Step 6: Substitute $B \cap X = \phi$ to get $B = B \cap A$ , proving $B \subset A$ .
Step 7: Conclude that since $A \subset B$ and $B \subset A$ , then $A = B$ .