Set Theory
Q1.
Let X be a set and 2^X denote the powerset of X. Define a binary operation \Delta on 2^X as follows:A \Delta B=(A-B) \cup (B-A) Let H=(2^X,\Delta ) . Which of the following statements about H is/are correct?Q2.
If A=\{x, y, z\} and B=\{u, v, w, x\}, and the universe is \{s, t, u, v, w, x, y, z\}. Then (A \cup \bar{B}) \cap(A \cap B) is equal toQ3.
Let S be a set of consisting of 10 elements. The number of tuples of the form (A,B) such that A and B are subsets of S, and A\subseteq B is _______Q4.
Let U=\{1,2,,...n\}. Let A=\{(x,X)|x\in X,X\subseteq U\}. Consider the following two statements on |A|. I. |A|=n2^{n-1}II. |A|=\sum_{k=1}^{n}k\binom{n}{k}Which of the above statements is/are TRUE?Q5.
The following paradigm can be used to find the solution of the problem in minimum time:Given a set of non-negative integer and a value K, determine if there is a subset of the given set with sum equal to K:Q6.
Consider the first order predicate formula: \forall x[\forall z\; z|x\Rightarrow ((z=x)\vee (z=1))\Rightarrow \exists w(w> x)\wedge (\forall z \; z|w\Rightarrow ((w=z)\vee (z=1)))] Here 'a|b' denotes that 'a divides b', where a and b are integers. Consider the following sets: S1: {1, 2, 3, ..., 100} S2: Set of all positive integers S3: Set of all integers Which of the above sets satisfy \varphi ?Q7.
Consider a set U of 23 different compounds in a Chemistry lab. There is a subset S of U of 9 compounds, each of which reacts with exactly 3 compounds of U. Consider the following statements: I. Each compound in U \ S reacts with an odd number of compounds. II. At least one compound in U \ S reacts with an odd number of compounds. III. Each compound in U n S reacts with an even number of compounds. Which one of the above statements is ALWAYS TRUE?Q8.
A function f:\mathbb{N}^{+}\rightarrow \mathbb{N}^{+}, defined on the set of positive integers \mathbb{N}^{+}, satisfies the following properties: f(n)=f(n/2) if n is even f(n)=f(n+5) if n is odd Let R={i|\exists j:f(j)=i} be the set of distinct values that f takes. The maximum possible size of R is ____.Q9.
Let N be the set of natural numbers. Consider the following sets. P: Set of Rational numbers (positive and negative) Q: Set of functions from {0, 1} to N R: Set of functions from N to {0, 1} S: Set of finite subsets of N. Which of the sets above are countable?Q10.
Let X and Y denote the sets containing 2 and 20 distinct objects respectively and F denote the set of all possible functions defined from X to Y. Let f be randomly chosen from F. The probability of f being one-to-one is ________.