Properties of crisp set: All at one place
Properties of the crisp set help us to simplify many mathematical set operations. Crisp sets are collections of unordered, district elements. We can perform various crisp set operations on the crisp set. It is recommended to the reader to first navigate through the crisp set operations for a better understanding of the properties of the crisp set.
Crisp set possesses the following properties. We will demonstrate each of them with a suitable example.
We will be using the following sets for further discussion:
X = {1, 2, 3, 4, 5, 6}
A = {1, 2, 3},
B = {2, 3, 4},
C = {5, 6}
Involution:
Involution states that the complement of complement of set A would be set A itself.
For the given data,
A’ = X – A = {4, 5, 6}
(A’)’ = X – A’ = {1, 2, 3} = A
Commutativity:
The commutativity property states that the operation can be performed irrespective of the order of the operand. For example, addition is a commutative operator, so 2 + 3 or 3 + 2 yields the same result. But, subtraction is not commutative, so 3 – 2 â 2 – 3.
Proving union is commutative:
A âĒ B = {1, 2, 3, 4} â LHS
B âĒ A = {1, 2, 3, 4} â RHS
Proving intersection is commutative:
A ⊠B = {2, 3} â LHS
B ⊠A = {2, 3} â RHS
Associativity:
The associativity property allows us to perform the operations by grouping the operands and keeping them in similar order.
(A âĒ B) âĒ C= A âĒ ( B âĒ C )
For given data:
A âĒ B = {1, 2, 3, 4}
(A âĒ B) âĒ C={1, 2, 3, 4, 5, 6} â LHS
B âĒ C = {2, 3, 4, 5, 6}
A âĒ (B âĒ C) = {1, 2, 3, 4, 5, 6} â RHS
(A ⊠B) ⊠C= A ⊠( B ⊠C)
For given data:
A ⊠B = {2, 3}
(A ⊠B) ⊠C = Ī â LHS
B ⊠C = Ī
A ⊠(B ⊠C) = Ī â RHS
Distributivity:
Mathematically it is defined as,
A âĒ (B ⊠C) = (A âĒ B) ⊠(A âĒ C)
B ⊠C = Ī
A âĒ (B ⊠C) = {1, 2, 3}âLHS
A âĒ B = {1, 2, 3, 4}
A âĒ C = {1, 2, 3, 5, 6}
(A âĒ B) ⊠(A âĒ C) = {1, 2, 3} â RHS
A ⊠(B âĒ C) = (A ⊠B) âĒ (A ⊠C)
B âĒ C = {2, 3, 4, 5, 6}
A ⊠(B âĒ C) = {2, 3} â LHS
A ⊠B = {2, 3}
A ⊠C = Ī
(A ⊠B) âĒ (A ⊠C) = {2, 3} â RHS
Absorption:
Mathematically absorption is defined as,
A âĒ (A ⊠B) = A
For the given data:
A ⊠B = {2, 3}
A âĒ (A ⊠B) = {1, 2, 3} = A
A ⊠(A âĒ B) = A
For the given data:
A âĒ B = {1, 2, 3, 4}
A ⊠(A âĒ B) = {1, 2, 3} = A
Idempotency/Tautology:
Idempotency is defined as,
A âĒ A = A
A ⊠A = A
For the given data,
A âĒ A = {1, 2, 3} = A
A ⊠A = {1, 2, 3} = A
Identity:
Mathematically, we can define this property as,
A âĒ X = X
A ⊠X = A
A âĒ Ī = A
A ⊠Ī = Ī
For the given data,
A âĒ X = {1, 2, 3, 4, 5, 6} = X
A ⊠X = {1, 2, 3} = A
A âĒ Ī = {1, 2, 3} = A
A ⊠Ī = { } = Ī
De Morgan’s Laws:
Mathematically, De Morgan’s laws are defined as,
Law 1: (A âĒ B)’ = A’ ⊠B’
For the given data:
A âĒ B = {1, 2, 3, 4}
(A âĒ B)’ = {5, 6} â LHS
A’ = {4, 5, 6}
B’ = {1, 5, 6}
A’ ⊠B’ = {5, 6} = (A âĒ B)’ â RHS
Law 2: (A ⊠B)’ = A’ âĒ B’
For the given data:
A ⊠B = {2, 3}
(A ⊠B)’ = {1, 4, 5, 6} â LHS
A’ = {4, 5, 6}
B’ = {1, 5, 6}
A’ âĒ B’ = {1, 4, 5, 6} = (A ⊠B)’ â RHS
Law of Contradiction:
Mathematically it is defined as,
A ⊠A’ = Ī
For the given data:
A’ = {4, 5, 6}
A ⊠A’ = { } = Ī
Law of Excluded Middle:
Mathematically it is defined as,
A âĒ A’ = X
For the given data:
A’ = {4, 5, 6}
A âĒ A’ = {1, 2, 3, 4, 5, 6} = X
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Test Your Knowledge:
Let X = {1, 2, 3, 4, 5, 6, 7, 8}, A = {1, 2, 3, 5}, B = {1, 3, 4, 6, 7}, C = {2, 4, 6, 8}. Prove the following properties for given sets.
- Involution
- Commutativity
- Associativity
- Distributivity
- Absorption
- Idempotency
- Identity
- De Morgan’s Laws
- Law of Contradiction
- Law of Excluded Middle
Please post your answer / query / feedback in comment section below !
lucidly explained.
Thanks a lot Mohmmad for interaction !