Identities Involving Operations in Sets Let Ø be the empty set, U be the universal set, A,B,C,X be any set. 1. XnØ=Ø 2. XnU= X 3. XUØ=X 4. XUU=U 5. (x°)° = X, the complement of the complement of a set is the set itself 6. (AUB)UC=AU(BUC) Associative property 7. AN(BNC)=(ANB)NC Associative property 8. AN(BUC)=(ANB)U(ANC) distributive property 9. AU(BNC)=(AUB)N(AUC) distributive property 10. (ANB)° = A° U B° De Morgan's Law %3D 11. (AUB)° = A°NB° De Morgan's Law 12. A-B = A N B° 13. AAB = (A-B) U (B-A) Exercises Draw a Venn diagram for each of the side (the right hand side and the left hand side of the equation) of the identities mentioned in the list of identities involving set operations. i. Let U= {1,2,3, . 20}, A={1,2, 3,4,5,6,7,8} , B= {2,3,4, 6,7,8,13,14,15}, C={2,4,6, ., 18,20}, and X= {10, 11, 12, ., 20}. Illustrate the identities involving operations in sets by solving the LHS and the RHS of the equations. ii.

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Set Theory

Every field in the present world involves a set. The theory of sets was developed by German mathematician Georg Cantor while working on the “problems of trigonometric series”.