Set Theory
Q31.
Let S={1,2,3....,m},m \gt 3. Let X_{1},...,X_{n} be subsets of S each of size 3. Define a function f from S to the set of natural numbers as, f(i) is the number of sets X_{j} that contain the element i. That is f(i)=|\{j|i\in X_{j}\}|. Then \sum_{i=1}^{m}f(i)Q32.
A binary operation \oplus on a set of integers is defined as x \oplus y= x^{2}+y^{2}. Which one of the following statements is TRUE about \oplus ?Q33.
Let A be a set with n elements. Let C be a collection of distinct subsets of A such that for any two subsets S_1 and S_2 in C, either S_1 \subset S_2 or S_2\subset S_1. What is the maximum cardinality of C?Q34.
Let A, B and C be non-empty sets and let X = (A - B) - C and Y = (A - C) - (B - C) Which one of the following is TRUE?Q35.
Consider the field C of complex numbers with addition and multiplication. Which of the following form(s) a subfield of C with addition and multiplication? S1: the set of real numbers S2:\{(a + ib) \mid a and b are rational numbers\} S3:\{a + ib \mid (a^2 + b^2) \leq 1\} S4: \{ia \mid a \text{ is real}\}Q36.
Let E,F and G be finite sets. Let X=(E \capF) - (F\capG) and Y = (E - (E\capG)) - (E - F). Which one of the following is true?Q37.
Let X, Y, Z be sets of sizes x, y and z respectively. Let W=X\timesY and E be the set of all subsets of W. The number of functions from Z to E isQ39.
A partial order \leq is defined on the set S=\left \{ x, a_1, a_2, \ldots, a_n, y \right \} \text{ as }x \leq _{i} a_{i} for all i and a_{i}\leq y for all i, where n \geq 1. The number of total orders on the set S which contain the partial order \leq isQ40.
Let n = p^{2}q, where p and q are distinct prime numbers. How many numbers m satisfy 1 \leq m \leq n and gcd(m,n)=1? Note that gcd(m,n) is the greatest common divisor of m and n.