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- 2. Introduction • The theory of sets was developed by German mathematician George Cantor. • A set is a collection of objects. • Objects in the collection are called elements of the set. • They are named by capital English alphabet.
- 3. Representation Of Sets• Roster form and Set Builder form • Roster Form- when the elements are written inside the set It is defined as a set by actually listing its elements, for example, the elements in the set A of letters of the English alphabet can be listed as A={a,b,c,……….,z} separated by comas.
- 4. • Set Builder Form- when we write a set in a straight form using underlying relations that binds them. • Example- {x | x < 6 and x is a counting number} in the set of all counting numbers less than 6. Note this is the same set as {1,2,3,4,5}.
- 5. Types Of Sets • Empty Sets • Finite Sets • Infinite Sets • Equal Sets • Subsets • Power Sets • Universal Sets
- 6. Empty Sets • A set that contains no members is called the empty set or null set . • For example, the set of the months of a year that have fewer than 15 days has no member .Therefore ,it is the empty set. The empty set is written as { } or .
- 7. Finite Sets • A set is finite if it consists of a definite number of different elements ,i.e., if in counting the different members of the set, the counting process can come to an end. • For example, if W be the set of people living in a town, then W is finite.
- 8. Infinite Sets • An infinite set is a set that is not a finite set. Infinite sets may be countable or uncountable. Some examples are: • The set of all integers, {..., -1, 0, 1, 2, ...}, is a count ably infinite set;
- 9. Equal Sets • Equal sets are sets which have the same members. Or Two sets a and b are said to be equal if they have the same no of elements. • For example, if P ={1,2,3},Q={2,1,3},R={3,2,1} then P=Q=R.
- 10. Subsets • Sets which are the part of another set are called subsets of the original set. • For example, if A={1,2,3,4} and B ={1,2} then B is a subset of A it is represented by .
- 11. Power Sets• If ‘A’ is any set then one set of all are subset of set ‘A’ that it is called a power set. • Example- If S is the set {x, y, z}, then the subsets of S are: • {} (also denoted , the empty set) • {x} • {y} • {z} • {x, y} • {x, z} • {y, z} • {x, y, z} • and hence the power set of S is {{}, {x}, {y}, {z}, {x, y}, {x, z}, {y, z}, {x, y, z}}.
- 12. Universal Sets • A universal set is a set which contains all objects, including itself. Or • In a group of sets if all the sets are the subset of a particular bigger set then that bigger set then that bigger set is called the universal set. • Example- A={12345678} B={1357} C={2468} D={2367} Here A is universal set and is denoted by
- 13. Operation Of Sets • Union of sets • Intersection of sets • Compliments of sets
- 14. Union • The union of two sets would be wrote as A U B, which is the set of elements that are members of A or B, or both too. • Using set-builder notation, A U B = {x : x is a member of A or X is a member of B}
- 15. Intersection • Intersection are written as A ∩ B, is the set of elements that are in A and B. • Using set-builder notation, it would look like: A ∩ B = {x : x is a member of A and x is a member of B}.
- 16. Complements • If A is any set which is the subset of a given universal set then its complement is the set which contains all the elements that are in but not in A. • Notation A’ ={1,2,3,4,5} A={1,2,3} A’={2,4}
- 17. Some Other Sets • Disjoint – If A ∩ B = 0, then A and B are disjoint. • Difference: B – A; all the elements in B but not in A • Equivalent sets – two sets are equivalent if n(A) = n(B).
- 18. Venn Diagrams • Venn diagrams are named after a English logician, John Venn. • It is a method of visualizing sets using various shapes. • These diagrams consist of rectangles and circles.