Let A, B and C be the sets such that \(A ∪ B = A ∪ C\) and \(A ∩ B = A ∩ C\). show that B = C.
Solution and Explanation
Let, A, B and C be the sets such that \(A ∪ B = A ∪ C\) and \(A ∩ B = A ∩ C.\)
To show: B = C
Let \(x ∈ B \)
\(⇒ x ∈ A ∪ B\space\space \space\space\space\space\space\space\space\space\space\space\space\space\space\space\space\space[ B ⊂ A ∪ B]\)
\(⇒ x ∈ A ∪ C\space\space\space\space\space\space\space\space\space\space\space\space\space\space\space\space\space\space\space [ A ∪ B = A ∪ C]\)
\(⇒ x ∈ A \space or ⇒ x ∈ C\)
Case I \(x ∈ A \)
Also, \(x ∈ B \)
\(∴ x ∈ A ∩ B\)
\(⇒ x ∈ A ∩ C [ ∴ A ∩ B = A ∩ C]\)
\(∴ x ∈ A\) and \(x ∈ C \)
\(∴ x ∈ C \)
\(∴ B ⊂ C \)
Similarly, we can show that \(C ⊂ B. \)
\(∴ B = C\)
Top Questions on Subsets
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P1: \( R \) is isomorphic to the product of two rings \( R_1 \) and \( R_2 \).
P2: \( \exists r_1, r_2 \in R \) such that \( r_1^2 = r_1 \neq 0 \), \( r_2^2 = 0 \), and \( r_1 + r_2 = 1 \).
P3: \( R \) has ideals \( I_1, I_2 \subset R \) with \( R \neq I_1 \), \( (0) \neq I_2 \), and \( R = I_1 + I_2 \) and \( I_1 \cap I_2 = (0) \).
P4: \( \exists a, b \in R \) with \( a \neq 0 \), \( b \neq 0 \) such that \( ab = 0 \).
Then, which of the following is/are TRUE? - Let \( X \) be an uncountable set. Let the topology on \( X \) be defined by declaring a subset \( U \subset X \) to be open if \( X - U \) is either empty or finite or countable, and the empty set to be open. Then, which of the following is/are TRUE?
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Concepts Used:
Operations on Sets
Some important operations on sets include union, intersection, difference, and the complement of a set, a brief explanation of operations on sets is as follows:
1. Union of Sets:
- The union of sets lists the elements in set A and set B or the elements in both set A and set B.
- For example, {3,4} ∪ {1, 4} = {1, 3, 4}
- It is denoted as “A U B”
2. Intersection of Sets:
- Intersection of sets lists the common elements in set A and B.
- For example, {3,4} ∪ {1, 4} = {4}
- It is denoted as “A ∩ B”
3.Set Difference:
- Set difference is the list of elements in set A which is not present in set B
- For example, {3,4} - {1, 4} = {3}
- It is denoted as “A - B”
4.Set Complement:
- The set complement is the list of all elements present in the Universal set except the elements present in set A
- It is denoted as “U-A”