Mock Test 1 — Sets

Let $A = \{x \in \mathbb{R} : x^2 - 3x + 2 = 0\}$ and $B = \{x \in \mathbb{R} : x^2 - x = 0\}$. The intersection set $A \cap B$ is:

If a finite set $S$ has exactly $n$ elements, the total number of subsets it can form is $2^n$. If a power set has $256$ elements, the cardinality of the original set is:

The simplified form of the complex set expression $(A \cup B) \cap (A \cup B')$ is:

According to De Morgan's dual identities, the complement expression $(A \cup B)'$ is completely identical to:

In a survey of $100$ students, $65$ watch football matches and $45$ watch cricket. If every student watches at least one sport, the number of students who watch both sports is:

Let $A = \{1, 2, \{3, 4\}\}$. Which of the following statements is mathematically true?

The interval difference set expression $(1, 4] - [3, 6)$ is equal to:

If $A$ and $B$ are two finite sets such that $A \subseteq B$, then the union operation $A \cup B$ always simplifies to:

The number of real solutions of the equation $x^2 + 4 = 0$ bounds the elements of a set. This set is classified as an:

The symmetric difference expression $A \triangle B$ can be rewritten as:

If the cardinality of set $A$ is $3$ and the cardinality of set $B$ is $4$, and the sets are completely disjoint ($A \cap B = \emptyset$), find the total number of elements inside the power set $|\mathcal{P}(A \cup B)|$.

In a competitive exam coaching cell, $40$ students registered for physics modules, $30$ for chemistry modules, and $15$ for both modules. Find the total number of students registered in the cell if every student takes at least one module.

Find the number of integer elements inside the set $X = \{x \in \mathbb{Z} : x^2 - 4 \le 0\}$.

Assertion (A): The power set of an empty set ($\mathcal{P}(\emptyset)$) is not empty; it contains exactly $1$ element.
Reason (R): The empty set $\emptyset$ is always a valid subset of any set, including itself, so it becomes an element inside the power set collection.

Assertion (A): For any two finite sets $A$ and $B$, the value of $|A - B|$ is given by $|A| - |B|$.
Reason (R): The set difference $A - B$ removes only the shared intersection elements from $A$, meaning the correct cardinality formula is $|A| - |A \cap B|$.