Question

Let A be the set of all subsets of a non-empty set S. Show that the binary operation 'union' on A is left distributive over the binary operation 'intersection' on A. Also show that 'intersection' is left distributive over 'union'.  

Answer 1

Let X, Y, Z ∈ A 

∴ By set theory, we have 

X ⋃ (Y ⋂ Z) = (X ⋃ Y)  (X ⋃ Z)which shows that 'union' is left distributive over 'intersection'. 

X ⋂ (Y ⋃ Z) = (X ⋂ Y) ⋃ (X ⋂ Z) which shows that 'intersection' is left distributive over 'union'.