List of top Mathematics 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?
Let \( E \subset F \) and \( F \subset K \) be field extensions which are not algebraic. Let \( \alpha \in K \) be algebraic over \( F \) and \( \alpha \notin F \). Let \( L \) be the subfield of \( K \) generated over \( E \) by the coefficients of the monic polynomial of minimal degree over \( F \) which has \( \alpha \) as a zero. Then, which of the following is/are TRUE?
The sum of the first \(n\) natural numbers is:
The number of subsets of the set A = {p,q} is
If A= {a, b, c, d} then number of subsets of A are
If A and B are d is joint sets, then
How many subsets the set P= (a, e, i, o, u) will have?
Make correct statements by filling in the symbols \( ⊂ \) or \(⊄\) in the blank spaces:
(i) {2, 3, 4} … {1, 2, 3, 4, 5}
(ii) {a, b, c} … {b, c, d}
(iii) {x: x is a student of Class XI of your school} … {x: x student of your school}
(iv) {x: x is a circle in the plane} … {x: x is a circle in the same plane with radius 1 unit}
(v) {x: x is a triangle in a plane}…{x: x is a rectangle in the plane}
(vi) {x: x is an equilateral triangle in a plane}… {x: x is a triangle in the same plane} 
(vii) {x: x is an even natural number} … {x: x is an integer}
Examine whether the following statements are true or false:
(i) {a, b} \(⊄\) {b, c, a}
(ii) {a, e} \(⊂\) {x: x is a vowel in the English alphabet}
(iii) {1, 2, 3} \(⊂\) {1, 3, 5}
(iv) {a} \(⊂ \) {a. b, c}
(v) {a} \(∈\) (a, b, c)
(vi) {x: x is an even natural number less than 6} \(⊂\) {x: x is a natural number which divides 36}
Write down all the subsets of the following sets:
(i) {a}
(ii) {a, b}
(iii) {1, 2, 3}
(iv) \(\phi\)
If A = {1, 2, 3, 4}, B = {3, 4, 5, 6}, C = {5, 6, 7, 8} and D = {7, 8, 9, 10}; find
(i) \(A∪ B\)
(ii) \(A ∪ C\)
(iii) \(B ∪ C\)
(iv) \(B ∪ D\)
(v) \(A ∪ B ∪ C\)
(vi) \(A∪ B ∪ D\)
(vii) \(B ∪ C ∪ D\)
Decide, among the following sets, which sets are subsets of one and another:
A = {x: x \(\in\) R and x satisfy \( x^2 - 8x + 12 = 0\)},
B = {2, 4, 6}, C = {2, 4, 6, 8…}, D = {6}.
Show that the following four conditions are equivalent: \((i) A ⊂ B (ii) A – B = \phi (iii) A ∪ B = B (iv) A ∩ B = A\)
Show that if \(A ⊂ B\), then \(C – B ⊂ C – A.\)
Show that for any sets A and B, \(A = (A ∩ B) ∪ (A – B)\) and \(A ∪ (B – A) = (A ∪ B)\)
Using properties of sets show that (i) \(A ∪ (A ∩ B) = A \) (ii) \(A ∩ (A ∪ B) = A.\)
Show that \(A ∩ B = A ∩ C\) need not imply B = C.
Let A and B be sets. If \(A ∩ X = B ∩ X = \phi\) and \(A ∪ X = B ∪ X\) for some set X, show that A = B.  (Hints \(A = A ∩ (A ∪ X), B = B ∩ (B ∪ X)\) and use distributive law)
Find sets A, B and C such that \(A ∩ B, B ∩ C\) and \(A ∩ C\) are non-empty sets and \(A ∩ B ∩ C = \phi.\)
Let A, B and C be the sets such that \(A ∪ B = A ∪ C\) and \(A ∩ B = A ∩ C\). show that B = C.