IIT JEE Main Maths -Unit 1- Union, intersection, and complement of sets - Study Notes-New Syllabus
IIT JEE Main Maths -Unit 1- Union, intersection, and complement of sets – Study Notes – New syllabus
IIT JEE Main Maths -Unit 1- Union, intersection, and complement of sets – Study Notes -IIT JEE Main Maths – per latest Syllabus.
Key Concepts:
- Union, Intersection and Complement of Sets
- Venn Diagrams and Their Operations
Union, Intersection and Complement of Sets
1. Union of Sets
The union of two sets \( A \) and \( B \) is the set of all elements which belong to \( A \) or \( B \) or both.
It is denoted by \( A \cup B \).
\( A \cup B = \{x \mid x \in A \text{ or } x \in B\} \)
2. Intersection of Sets
The intersection of two sets \( A \) and \( B \) is the set of all elements common to both.
It is denoted by \( A \cap B \).
\( A \cap B = \{x \mid x \in A \text{ and } x \in B\} \)
3. Complement of a Set
The complement of a set \( A \) (with respect to universal set \( U \)) is the set of all elements in \( U \) that are not in \( A \). It is denoted by \( A’ \) or \( A^{c} \).
\( A’ = \{x \in U \mid x \notin A\} \)
Example
If \( A = \{1, 2, 3\} \) and \( B = \{3, 4, 5\} \), find \( A \cup B \).
▶️ Answer / Explanation
\( A \cup B = \{1, 2, 3, 4, 5\} \)
Explanation: The union includes every distinct element that appears in either \( A \) or \( B \).
Example
If \( A = \{2, 4, 6, 8\} \) and \( B = \{4, 8, 10, 12\} \), find \( A \cap B \).
▶️ Answer / Explanation
\( A \cap B = \{4, 8\} \)
Explanation: Only elements common to both sets are included in the intersection.
Example
If the universal set \( U = \{1, 2, 3, 4, 5, 6, 7\} \) and \( A = \{2, 4, 6\} \), find \( A’ \).
▶️ Answer / Explanation
\( A’ = \{1, 3, 5, 7\} \)
Explanation: The complement includes all elements of \( U \) that are not in \( A \).
Venn Diagrams and Their Operations
A Venn diagram is a visual representation of sets using closed curves (usually circles). It helps illustrate relationships such as union, intersection, and complement among sets.
Let \( U \) be the universal set that contains all possible elements under consideration. Subsets of \( U \) are represented by circles within a rectangle representing \( U \).
Common Set Operations Using Venn Diagrams:
Union (\( A \cup B \)): The set of all elements that belong to \( A \), or \( B \), or both.
Venn region: The combined area of both circles.
Symbolically: \( A \cup B = \{x \mid x \in A \text{ or } x \in B\} \)
Intersection (\( A \cap B \)): The set of all elements common to both \( A \) and \( B \).
Venn region: The overlapping area of the two circles.
Symbolically: \( A \cap B = \{x \mid x \in A \text{ and } x \in B\} \)
Difference (\( A – B \)): The set of elements that belong to \( A \) but not to \( B \).
Venn region: Portion of circle \( A \) excluding the overlap.
Symbolically: \( A – B = \{x \mid x \in A \text{ and } x \notin B\} \)
Complement (\( A’ \)): The set of all elements of the universal set \( U \) that are not in \( A \).
Venn region: The region outside circle \( A \).
Symbolically: \( A’ = \{x \mid x \in U, \, x \notin A\} \)
Important Identities:
- \( (A’)’ = A \)
- \( A \cup A’ = U \)
- \( A \cap A’ = \varnothing \)
- \( A \cup \varnothing = A \), \( A \cap \varnothing = \varnothing \)
- \( A \cup U = U \), \( A \cap U = A \)
Example
Let \( U = \{1, 2, 3, 4, 5, 6, 7, 8\} \), \( A = \{2, 4, 6, 8\} \), and \( B = \{1, 2, 3, 4\} \). Find \( A \cup B \), \( A \cap B \), \( A – B \), and \( A’ \).
▶️ Answer / Explanation
Step 1: Union:
\( A \cup B = \{1, 2, 3, 4, 6, 8\} \)
Step 2: Intersection:
\( A \cap B = \{2, 4\} \)
Step 3: Difference:
\( A – B = \{6, 8\} \)
Step 4: Complement of \( A \) (with respect to \( U \)):
\( A’ = \{1, 3, 5, 7\} \)
Step 5: Verification:
Each operation can be verified easily using a Venn diagram by shading the respective regions.