Set Theory

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 _______

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 to

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 ?

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?

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:

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?

The symmetric difference of sets A={1,2,3,4,5,6,7,8} and B={1,3,5,6,7,8,9} is:

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?

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 ____.

Suppose U is the power set of the set S={1,2,3,4,5,6}. For any T\inU, let |T| denote the number of elements in T and T' denote the complement of T. For any T,R\inU, let T\R be the set of all elements in T which are not in R. Which one of the following is true?