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Example 32 - Let P be set of all subsets of a given set X - Examples

Example 32  - Chapter 1 Class 12 Relation and Functions - Part 2
Example 32  - Chapter 1 Class 12 Relation and Functions - Part 3 Example 32  - Chapter 1 Class 12 Relation and Functions - Part 4

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Transcript

Question 15(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 , {1}, {2}, {3}, {1, 2}, {1, 3} , {2, 3} , {1, 2, 3} P is a set of all subsets of X. Hence P = { , {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

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Davneet Singh

Davneet Singh has done his B.Tech from Indian Institute of Technology, Kanpur. He has been teaching from the past 13 years. He provides courses for Maths, Science, Social Science, Physics, Chemistry, Computer Science at Teachoo.