NCERT Solutions & Concepts for Class 11 Maths Chapter 1 Exercise 1.5

Understanding the Complement of a Set

To understand the complement of a set, we must first establish a reference boundary known as the Universal Set (denoted by $U$). Let $A$ be a subset of $U$. The complement of $A$ (denoted by $A'$ or $A^c$) is the set of all elements belonging to $U$ that do not belong to $A$.

Mathematically, we write this as:
$A' = \{x \in U : x \notin A\}$

In simple terms, you can calculate the complement using the set difference operation: $A' = U - A$. Visually, if you picture a Venn diagram where the rectangle represents $U$ and a circle inside represents $A$, the shaded region outside the circle $A$ but inside the rectangle is $A'$. This basic concept extends to complex relations, especially when verifying De Morgan's Laws.

Properties of Complement Sets & De Morgan's Laws

Complement Laws

1. $A \cup A' = U$ (The union of a set and its complement equals the Universal Set). 2. $A \cap A' = \phi$ (A set and its complement share no common elements; they are disjoint).

De Morgan's Laws

These are highly crucial equations for boards: 1. $(A \cup B)' = A' \cap B'$ (The complement of union is the intersection of complements). 2. $(A \cap B)' = A' \cup B'$ (The complement of intersection is the union of complements).

Law of Double Complementation

$(A')' = A$. Taking the complement of a complement set returns you back to the original set.

Laws of Empty Set & Universal Set

$\phi' = U$ (Complement of nothing is everything) and $U' = \phi$ (Complement of everything is nothing).

How to Find and Verify Complement Set Operations

1. Identify the Universal Set — Always determine the scope of elements listed in the Universal Set ($U$). All operations are confined to this set.
2. Perform the Inner Operation First — If asked to find $(A \cup B)'$, first calculate the union set $A \cup B$ by listing all unique elements in both $A$ and $B$.
3. Subtract from Universal Set — Remove the elements of the inner operation from $U$. The remaining elements in $U$ form the final complement set.

Fully Worked CBSE Examples

• Example 1: Let $U = \{1, 2, 3, 4, 5, 6, 7, 8, 9\}$, $A = \{2, 4, 6, 8\}$, and $B = \{2, 3, 5, 7\}$. Find $(A \cup B)'$. Step 1: First find $A \cup B$. Combining elements from both sets gives $A \cup B = \{2, 3, 4, 5, 6, 7, 8\}$. Step 2: Now, find the complement of $(A \cup B)$ with respect to $U$. Subtract these elements from $U$. $(A \cup B)' = U - \{2, 3, 4, 5, 6, 7, 8\} = \{1, 9\}$.
• Example 2: Using the same sets, verify De Morgan's Law: $(A \cap B)' = A' \cup B'$. Step 1: Find the Left Hand Side (LHS): $A \cap B = \{2\}$ (since only 2 is common). $(A \cap B)' = U - \{2\} = \{1, 3, 4, 5, 6, 7, 8, 9\}$. Step 2: Find the Right Hand Side (RHS): $A' = U - A = \{1, 3, 5, 7, 9\}$ $B' = U - B = \{1, 4, 6, 8, 9\}$ $A' \cup B' = \{1, 3, 4, 5, 6, 7, 8, 9\}$. Since LHS = RHS, the law is verified!

Board Exam Trap: Watch out for parentheses and operators!

A very common error Class 11 students commit during exams is writing $(A \cup B)' = A' \cup B'$. Remember, the union symbol flips to intersection, and intersection flips to union when distributing the complement operator.
Always verify your operations using a quick Venn Diagram sketch in your rough work section to ensure you haven't made silly computational slips!

Practice Questions with Solutions

• Q: Let $U = \{x : x \in \mathbb{N}, x \le 10\}$ and $A = \{x : x \text{ is a prime number} \}$. Find $A'$. A: Step 1: Write down the sets in roster form: $U = \{1, 2, 3, 4, 5, 6, 7, 8, 9, 10\}$ $A = \{2, 3, 5, 7\}$ Step 2: Calculate $A' = U - A$ by removing the elements of $A$ from $U$. Final answer: $A' = \{1, 4, 6, 8, 9, 10\}$
• Q: Let $U$ be the set of all triangles in a plane. If $A$ is the set of all triangles with at least one angle different from $60^\circ$, what is $A'$? A: Step 1: Analyze the definition of set $A$. If a triangle has at least one angle different from $60^\circ$, it means it is NOT equilateral (since equilateral triangles have all angles equal to $60^\circ$). Step 2: The complement set $A'$ contains triangles from $U$ that do not fall under $A$. Therefore, $A'$ must contain triangles where NO angle is different from $60^\circ$. Final answer: $A'$ is the set of all equilateral triangles.
• Q: Let $U = \{1, 2, 3, 4, 5, 6, 7, 8, 9, 10\}$, $A = \{1, 3, 5, 7, 9\}$, and $B = \{2, 4, 6, 8, 10\}$. Find $(A \cup B)'$. A: Step 1: Calculate $A \cup B$. Since $A$ is odd natural numbers up to 10 and $B$ is even natural numbers up to 10: $A \cup B = \{1, 2, 3, 4, 5, 6, 7, 8, 9, 10\} = U$ Step 2: Find the complement: $(A \cup B)' = U - U = \phi$ Final answer: $\phi$ (Empty set)
• Q: Simplify the expression: $(A \cup B)' \cup (A' \cap B)$ using set laws. A: Step 1: Apply De Morgan's Law to the first term: $(A \cup B)' = A' \cap B'$ Step 2: Substitute this back into the expression: $(A' \cap B') \cup (A' \cap B)$ Step 3: Apply the Distributive Law in reverse (take $A' \cap$ common): $= A' \cap (B' \cup B)$ Step 4: Use Complement Law ($B' \cup B = U$): $= A' \cap U$ Step 5: Intersection with universal set yields the set itself: $= A'$ Final answer: $A'$

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