Answer to Question #131515 in Abstract Algebra for Joseph Se

Question

Let X be a non-empty set. Define the set Z to be the collection of all subsets of X. Consider the binary operation “union” in the set Z, that is, for A and B that are subsets of X, let
A*B=A∪B

Determine which of the properties of the binary operation is/are satisfied in Z under *.

Answer 1

$$Solution$$

a. ) Commutative Properties:  The Commutative Property for Union and the Commutative Property for binary operation (*) says that the order of the sets in which we do the operation does not change the result.

$A\ *\ B=A\ \cup\ B,\ \forall\ \ A\ ,B\ \in\ Z\ -----\ equation\ (I)\\ But\ A\ \cdot\ B=A\ \cup\ B\ \forall\ A,\ B\in\ Z\\ a.\ Commutative\ Law\\ A \cdot B=B \cdot A\\ LHS\implies\ A \cdot B=A\ \cup B=B\ \cup\ A=B \cdot A\\ \therefore\ LHS=RHS\\ Hence\ Z\ follows\ binary\ property\ of\ commutative\ property\\$

b. ) Associative Properties:  The Associative Property for Union and the Commutative Property for binary operation (*) says that how the sets are grouped does not change the result.

$General\ property:\ (A \cdot B) \cdot C=A \cdot (B \cdot C)\\ LHS\\ \implies\ (A \cdot B) \cdot C=(A \cup B) \cup C\\ Now\ RHS\implies\ A \cdot (B \cdot C)=A \cup (B \cup C)\\ As\ per\ equation\ (I)\,\\ =A \cup (B \cup C)=A \cdot(B \cdot C)\\ Hence\ Z\ follows\ binary\ property\ of\ Associative\ property\\$

c. ) Distributive Properties: The Distributive Property of Union over the binary operation * show two ways of finding results for certain problems mixing the set operations of union binary operation (*)

$General\ property:\ A \cup (B \cdot C)= (A \cup B) \cdot (A \cup C)\ and\ A \cdot (B \cup C)= (A \cdot B) \cup (A \cdot C)\\ Hence\ Z\ follows\ binary\ property\ of\ Distributive\ property.\\$

Hence proof: Under Z it will follow commutative, associative and distributive properties.