Example 32 - Class 12

Transcript

Example 32 (Introduction) Let P be the set of all subsets of a given set X. Show that ∪: P × P→P given by (A, B) →A ∪ B and ∩: P × P →P given by (A, B) →A ∩ B are binary operations on the set P Let X = {1, 2, 3} Subsets of X are ϕ , {1}, {2}, {3}, {1, 2}, {1, 3} , {2, 3} , {1, 2, 3} P is a set of all subsets of X. Hence P = { ϕ , {1}, {2}, {3}, {1, 2}, {1, 3} , {2, 3} , {1, 2, 3} } If we take union of any 2 elements of P and calculate its union, it will always fall in P For example Taking 2 elements {1} and {1, 2} ( {1}, {1, 2}) → {1} ∪ {1, 2} → {1, 2} This {1, 2} is in P So, ∪ is a binary operation Similarly, for intersection(∩) ( {1}, {1, 2}) → {1} ∩ {1, 2} → {1} This {1, 2} is in P So, ∩ is a binary operation Lets prove it generally Example 32 Let P be the set of all subsets of a given set X. Show that ∪: P × P→P given by (A, B) →A ∪ B and ∩: P × P →P given by (A, B) →A ∩ B are binary operations on the set P Union ∪: P × P → P (A, B) →A ∪ B P is a the set of all subsets of a given set X Here, A & B are in set P, hence we can say that A and B are also subsets of X If we calculate A ∪ B A ∪ B will be a subset of X as union of subsets is also a subset Hence, A ∪ B will also be in set P So, ∪ is a binary operation Intersection ∩ : P × P → P (A, B) →A ∩ B P is a the set of all subsets of a given set X Here, A & B are in set P, hence we can say that A and B are also subsets of X If we calculate A ∩ B A ∩ B will be a subset of X as union of subsets is also a subset Hence, A ∩ B will also be in set P So, ∩ is a binary operation