In simple terms, these functions and laws describe how operations such as intersection, union, and complement work on sets, as well as the fundamental rules governing these operations. For instance, the intersection operation identifies the common elements between two sets, while the union operation combines all elements from both sets. The complement operation determines the elements not present in a given set. Let us discuss these concepts in more detail below.

1.2.1. Commutative Laws

a. ( A ∪ B = B ∪ A )
The union of two sets is the same regardless of the order of the sets. For example, ( A ∪ B ) and ( B ∪ A ) yield the same set.

b. ( A ∩ B = B ∩ A )
The intersection of two sets remains the same even if the order is reversed. In other words, ( A ∩ B ) and ( B ∩ A ) are identical.

1.2.2. Associative Laws

a. A∪(B∪C) = (A∪B)∪C
In the case of the union operation, the order in which sets are grouped does not affect the result. For example, merging BBand CC first and then combining the result with AA yields the same outcome as merging AA and BB first and then adding CC.

b. A∩(B∩C) = (A∩B)∩C
Similarly, for the intersection operation, the order of grouping sets does not change the result.

1.2.3. Distributive Laws

a. ( A ∩ (B ∪ C) = (A ∩ B) ∪ (A ∩ C) )
The intersection of set ( A ) with ( B ∪ C ) is equal to the union of the intersections of ( A ) with ( B ) and ( A ) with ( C ).

b. ( A ∪ (B ∩ C) = (A ∪ B) ∩ (A ∪ C) )
The union of set ( A ) with ( B ∩ C ) is equal to the intersection of the union of ( A ) and ( B ) with the union of ( A ) and ( C ).

1.2.4. Identity Laws

a. ( A ∪ ∅ = A )
The union of a set with the empty set is the set itself. Since the empty set contains no elements, it does not affect the union.

b. ( A ∩ E = A )
The intersection of a set with the universal set is the set itself, as the universal set contains all possible elements.

1.2.5. Complement Laws

a. ( A ∪ A′ = E )
The union of a set and its complement (the elements not in the set) results in the universal set.

b. ( A ∩ A′ = ∅ )
The intersection of a set and its complement is the empty set, as they have no elements in common.

1.2.6. Idempotent Laws

a. ( A ∪ A = A )
The union of a set with itself yields the same set.

b. ( A ∩ A = A )
The intersection of a set with itself yields the same set.