Show that the following four conditions are equivalent: \((i) A ⊂ B (ii) A – B = \phi (iii) A ∪ B = B (iv) A ∩ B = A\)
Solution and Explanation
First, we have to show that \((i) ⇔ (ii).\)
Let \(A ⊂ B\)
To show: \(A – B \ne\phi\)
If possible, suppose \(A – B \ne\phi\)
This means that there exists \(x ∈ A, x ≠ B\), which is not possible as \(A ⊂ B.\)
\(∴ A – B = \phi\)
\(∴ A ⊂ B ⇒ A – B = \phi\)
Let \(A – B = \phi\)
To show: \(A ⊂ B\)
Let \(x ∈ A\)
Clearly, \(x ∈ B\) because if \(x ∉ B\), then \(A – B ≠ \phi\)
\(∴ A – B = \phi ⇒ A ⊂ B\)
\(∴ (i) ⇔ (ii)\)
Let \(A ⊂ B\)
To show: \(A ∪ B = B\)
Clearly, \(B ⊂ A ∪ B\)
Let \(x ∈ A ∪ B\)
\(⇒ x ∈ A \space or x ∈ B\)
Case I: \(x ∈ A\)
\(⇒ x ∈ B [∴ A ⊂ B]\)
\(∴ A ∪ B ⊂ B\)
Case II: \(x ∈ B\)
Then, \(A ∪ B = B\)
Conversely, let \(A ∪ B = B\)
Let \(x ∈ A\)
\(⇒ x ∈ A ∪ B\) \([∴ A ⊂ A ∪ B]\)
\(⇒ x ∈ B\) \([ ∴ A ∪ B = B]\)
\(∴ A ⊂ B\)
Hence, \((i) ⇔ (iii)\)
Now, we have to show that \((i) ⇔ (iv).\)
Let \(A ⊂ B\)
Clearly \(A ∩ B ⊂ A\)
Let \(x ∈ A\)
We have to show that \(x ∈ A ∩ B\)
As \(A ⊂ B, x ∈ B\)
\(∴ x ∈ A ∩ B\)
\(∴ A ⊂ A ∩ B\)
Hence, \(A = A ∩ B\)
Conversely, suppose. \(A ∩ B = A\)
Let \(x ∈ A\)
\(⇒ x ∈ A ∩ B\)
\(⇒ x ∈ A\) and \(x ∈ B\)
\(⇒ x ∈ B\)
\(∴ A ⊂ B\)
Hence, \((i) ⇔ (iv).\)
Top Questions on Subsets
Consider the following subsets of the Euclidean space \( \mathbb{R}^4 \):
\( S = \{ (x_1, x_2, x_3, x_4) \in \mathbb{R}^4 : x_1^2 + x_2^2 + x_3^2 - x_4^2 = 0 \} \),
\( T = \{ (x_1, x_2, x_3, x_4) \in \mathbb{R}^4 : x_1^2 + x_2^2 + x_3^2 - x_4^2 = 1 \} \),
\( U = \{ (x_1, x_2, x_3, x_4) \in \mathbb{R}^4 : x_1^2 + x_2^2 + x_3^2 - x_4^2 = -1 \} \).
Then, which one of the following is TRUE?Let the functions \( f: \mathbb{R} \to \mathbb{R} \) and \( g: \mathbb{R}^2 \to \mathbb{R} \) be given by \[ f(x_1, x_2) = x_1^2 + x_2^2 - 2x_1x_2, \quad g(x_1, x_2) = 2x_1^2 + 2x_2^2 - x_1x_2. \] Consider the following statements:
S1: For every compact subset \( K \) of \( \mathbb{R} \), \( f^{-1}(K) \) is compact.
S2: For every compact subset \( K \) of \( \mathbb{R} \), \( g^{-1}(K) \) is compact. Then, which one of the following is correct?- In the following, all subsets of Euclidean spaces are considered with the respective subspace topologies. Define an equivalence relation \( \sim \) on the sphere \[ S = \left\{ (x_1, x_2, x_3) \in \mathbb{R}^3 : x_1^2 + x_2^2 + x_3^2 = 1 \right\} \] by \( (x_1, x_2, x_3) \sim (y_1, y_2, y_3) \) if \( x_3 = y_3 \), for \( (x_1, x_2, x_3), (y_1, y_2, y_3) \in S \). Let \( [x_1, x_2, x_3] \) denote the equivalence class of \( (x_1, x_2, x_3) \), and let \( X \) denote the set of all such equivalence classes. Let \( L : S \to X \) be given by \[ L\left( (x_1, x_2, x_3) \right) = [x_1, x_2, x_3]. \] If \( X \) is provided with the quotient topology induced by the map \( L \), then which one of the following is 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?
- All rings considered below are assumed to be associative and commutative with \( 1 \neq 0 \). Further, all ring homomorphisms map 1 to 1.
Consider the following statements about such a ring \( R \):
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?
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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”