Below is a clear, complete, and exam-oriented explanation of Combination of Sets in Discrete Structures.
⭐ Combination of Sets
“Combination of Sets” refers to the different operations we can perform on sets to combine, relate, or modify them. These operations help us understand how different sets interact with each other.
The most important combination operations are:
- Union
- Intersection
- Difference
- Symmetric Difference
- Complement
- Cartesian Product
Let’s discuss each in detail.
1️⃣ Union of Sets (A ∪ B)
Definition
The union of two sets A and B contains all elements that belong to A or B or both.
Notation:
( A \cup B )
Example:
A = {1, 2, 3}
B = {3, 4, 5}
( A \cup B = {1, 2, 3, 4, 5} )
Venn Diagram Idea:
Shaded region includes both circles.
2️⃣ Intersection of Sets (A ∩ B)
Definition
The intersection of two sets A and B contains the elements that belong to both A and B.
Notation:
( A \cap B )
Example:
A = {1, 2, 3}
B = {3, 4, 5}
( A \cap B = {3} )
3️⃣ Difference of Sets (A – B or A \ B)
Definition
The difference of sets contains elements that are in A but not in B.
Notation:
( A – B )
Example:
A = {1, 2, 3, 4}
B = {3, 4, 5}
( A – B = {1, 2} )
Note:
Difference is not symmetric:
( A – B \neq B – A )
4️⃣ Symmetric Difference (A Δ B)
Definition
Symmetric difference contains elements that belong to either A or B, but not both.
Notation:
( A \Delta B )
Example:
A = {1, 2, 3}
B = {3, 4, 5}
( A \Delta B = {1, 2, 4, 5} )
Formula:
( A \Delta B = (A – B) \cup (B – A) )
5️⃣ Complement of a Set (A′)
Definition
The complement of A contains all elements in the universal set (U) that are not in A.
Notation:
A′ or ( A^c )
Example:
Let U = {1, 2, 3, 4, 5}
A = {2, 3}
A′ = {1, 4, 5}
6️⃣ Cartesian Product (A × B)
Definition
Cartesian product gives a set of ordered pairs, where the first element is from A and the second from B.
Notation:
( A \times B )
Example:
A = {1, 2}, B = {x, y}
( A \times B = {(1, x), (1, y), (2, x), (2, y)} )
Used in — relations, functions, database tables, graph theory.
⭐ Important Properties of Combination of Sets
Union
- Commutative: ( A \cup B = B \cup A )
- Associative: ( (A \cup B) \cup C = A \cup (B \cup C) )
Intersection
- Commutative: ( A \cap B = B \cap A )
- Associative: ( (A \cap B) \cap C = A \cap (B \cap C) )
Distributive Laws
- ( A \cap (B \cup C) = (A \cap B) \cup (A \cap C) )
- ( A \cup (B \cap C) = (A \cup B) \cap (A \cup C) )
De Morgan’s Laws
- ( (A \cup B)’ = A’ \cap B’ )
- ( (A \cap B)’ = A’ \cup B’ )