How to Solve Union and Intersection Problems in Sets

The resolution of practical problems involving the union and intersection of two sets requires systematic application of set-theoretic results, use of standard notations, and rigorous attention to conditions and constraints relevant to JEE Main-level questions.

Formal Representation of Union and Intersection of Two Sets

Definition: For any sets $A$ and $B$ in a universal set $U$, the union $A \cup B$ is defined as $\{ x \in U: x \in A \text{ or } x \in B \}$, and the intersection $A \cap B$ is $\{ x \in U: x \in A \text{ and } x \in B \}$.

The set difference $A \setminus B$ denotes elements in $A$ but not in $B$. The complement of $A$ is $A' = U \setminus A$. These concepts are foundational for the algebraic treatment of practical set problems.

Standard Result: Counting Elements Using Union and Intersection

Result: For finite sets $A$ and $B$, $n(A \cup B) = n(A) + n(B) - n(A \cap B)$, where $n(A)$ denotes the number of elements in $A$.

This result is derived directly from the principle of inclusion-exclusion for two sets and is frequently applied in problems involving cardinalities and survey problems.

Problem Categories and Systematic Approaches

• Direct computation of $A \cup B$, $A \cap B$ from listed sets
• Determination of unknowns in cardinality equations
• Survey-type word problems (students, voters, objects)
• Verification of set identities using specific sets

Solved Problems Involving Union and Intersection of Two Sets

Example: Let $A = \{2,4,6,8\}$, $B = \{4,5,6,7\}$. Compute $A \cup B$ and $A \cap B$.

$A \cap B$ consists of elements common to both sets: $A \cap B = \{4,6\}$.

$A \cup B$ consists of all elements from both sets, without repetition: $A \cup B = \{2,4,5,6,7,8\}$.

Example: In a class of 40 students, 25 like Mathematics, 18 like Physics, and 10 like both subjects. How many students like at least one of the two subjects?

Given $n(A) = 25$, $n(B) = 18$, $n(A \cap B) = 10$.

Apply the counting result: $n(A \cup B) = 25 + 18 - 10 = 33$.

Thus, 33 students like at least one of Mathematics or Physics.

Example: Determine $A \cup B$ and $A \cap B$ when $A = \{ x : x \text{ is an even integer, } 2 \leq x \leq 10 \}$, $B = \{ x : x \text{ is a prime number, } 1 \leq x \leq 10 \}$.

Explicitly, $A = \{2,4,6,8,10\}$, $B = \{2,3,5,7\}$.

$A \cap B = \{2\}$; $A \cup B = \{2,3,4,5,6,7,8,10\}$.

Algebraic Verification of Inclusion-Exclusion Formula

Exam Tip: Always verify the application of $n(A \cup B) = n(A) + n(B) - n(A \cap B)$ by listing elements when possible, especially in cases where $A$ and $B$ are described in roster or set-builder forms.

Constraints in Problems Involving Union and Intersection

• Empty intersection (disjoint sets)
• Non-empty intersection/subsets scenarios
• Relation to universal set and complements
• Application to finite and infinite sets distinctly

For further detail on principles discussed, refer to Union, Intersection, and Difference of Sets.

Application to Survey Problems and Unknown Quantity Determination

If $n(A \cup B)$ and $n(A \cap B)$ are given in context-based questions, the formula rearranges to yield unknowns; for example, $n(A) = n(A \cup B) - n(B) + n(A \cap B)$.

When additional conditions are present (such as students liking neither of the activities), introduce variable $x = \text{number liking neither}$, and write $n(U) = n(A \cup B) + x$ if $U$ is the universal set of students.

Frequent Error Patterns in Set Operation Problems

• Double-counting elements in $A \cap B$ when listing $A \cup B$
• Ignoring empty intersections when sets are disjoint
• Misassignment of formula to complements