The laws discussed here are derived from the fundamental set laws mentioned in Section 1.2 and illustrate how set operations can be manipulated at a more advanced level. In this section, we will examine these laws in detail to better understand their applications and implications in set theory.

1.3.1. Idempotent Laws

a. (A∩B) ∪ (A∩B)= A∩B
The union of a set’s intersection with itself yields the same intersection. In other words, the union of a set with itself results in the set itself.

b. (A∪B)∩(A∪B)=A∪B
Similarly, the intersection of a set’s union with itself results in the same union. That is, the intersection of a set with itself gives the set itself.

1.3.2. Absorption Laws

a. A∪(A∩B)=A
This law states that the union of set AA with the intersection of A and B results in A itself. Logically, since A∩B is a subset of A, its union with A does not alter A.

b. A∩(A∪B)=A
This law states that the intersection of set AA with the union of AA and BB results in AA itself. Since AA is a subset of A∪BA∪B, their intersection does not change AA.

1.3.3. Minimisation Laws

A∪∅=A  and A∩E=A
These laws express the minimisation in set operations. The union of a set with the empty set yields the set itself (as the empty set contributes nothing to the union). Similarly, the intersection of a set with the universal set results in the set itself (as the universal set contains all elements).

1.3.4. De Morgan’s Laws

a. (A∪B)′=A′∩B′
The complement of a union is equal to the intersection of the complements of the individual sets. In other words, the complement of AA union BB is the same as the intersection of the complements of AA and BB.

b. (A∩B)′=A′∪B′
The complement of an intersection is equal to the union of the complements of the individual sets. Thus, the complement of AA intersection BB is the same as the union of the complements of AA and BB.

De Morgan’s laws provide an important rule for the complement and distribution of operations on sets. They are particularly useful for simplifying complex set expressions and for transforming logical operations. These laws form a fundamental part of set theory and logical calculations and can be used to solve more complex problems.