GATE CS Sets, Relations and Functions questions

A polynomial $p(x)$ satisfies the following: $p(1) = p(3) = p(5) = 1$ $p(2) = p(4) = -1$ The minimum degree of such a polynomial is

Let $L$ be a set with a relation $R$ which is transitive, anti-symmetric and reflexive and for any two elements $a, b \in L$, let the least upper bound $lub (a, b)$ and the greatest lower bound $glb (a, b)$ exist. Which of the following is/are true?

The number of binary relations on a set with $n$ elements is:

In a room containing $28$ people, there are $18$ people who speak English, $15$ people who speak Hindi and $22$ people who speak Kannada. $9$ persons speak both English and Hindi, $11$ persons speak both Hindi and Kannada whereas $13$ persons speak both Kannada and English. How many speak all three languages?

The binary relation $R = \{(1, 1), (2, 1), (2, 2), (2, 3), (2, 4), (3, 1), (3, 2), (3, 3), (3, 4)\}$ on the set $A=\{1, 2, 3, 4\}$ is

The number of functions from an $m$ element set to an $n$ element set is

Let $R_1$ and $R_2$ be two equivalence relations on a set. Consider the following assertions: $R_1 \cup R_2$ is an equivalence relation $R_1 \cap R_2$ is an equivalence relation Which of the following is correct?

Suppose $A$ is a finite set with $n$ elements. The number of elements in the largest equivalence relation of A is

A polynomial $p(x)$ is such that $p(0) = 5, p(1) = 4, p(2) = 9$ and $p(3) = 20$. The minimum degree it should have is

Let $R$ be a non-empty relation on a collection of sets defined by $_{A}R_ B$ if and only if $A \cap B = \phi$. Then, (pick the true statement)

Let $R$ denote the set of real numbers. Let $f:R\times R \rightarrow R \times R$ be a bijective function defined by $f(x,y) = (x+y, x-y)$. The inverse function of $f$ is given by

Suppose $X$ and $Y$ are sets and $|X| \text{ and } |Y|$ are their respective cardinality. It is given that there are exactly $97$ functions from $X$ to $Y$. From this one can conclude that

Let $A$ and $B$ be sets and let $A^c$ and $B^c$ denote the complements of the sets $A$ and $B$. The set $(A-B) \cup (B-A) \cup (A \cap B)$ is equal to

Determine the number of positive integers $(\leq 720)$ which are not divisible by any of $2,3$ or $5.$

Determine the number of divisors of $600.$

The number of elements in the power set $P(S)$ of the set $S=\{\{\emptyset\}, 1, \{2, 3\}\}$ is:

Let $R$ be a symmetric and transitive relation on a set $A$. Then

Every subset of a countable set is countable. State whether the above statement is true or false with reason.

Let $A$ and $B$ be sets with cardinalities $m$ and $n$ respectively. The number of one-one mappings from $A$ to $B$, when $m < n$, is

Let $S$ be an infinite set and $S_1 \dots , S_n$ be sets such that $S_1 \cup S_2 \cup \dots \cup S_n = S$. Then