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SETS.

- 1. A = {1, 3, 2, 5} n(A) = 4 Sets use “curly” brackets The number of elements in Set A is 4 Sets are denoted by Capital letters A3∈ A7∉ 3 is an element of A 7 is not an element of A SETS A set is a collection of well defined distinct objects. The objects of the set are called elements.
- 2. {1, 3, 2, 3, 5, 2} We never repeat elements in a set.{1, 2, 3, 5} Sets can be represented by two methods: Rooster or Tabular form: In this the elements are separated by commas. e.g. set A of all odd natural numbers less than 10. A = {1,3,5,7,9} Set builder method: In this the common property of the elements is specified. e.g. set A of all odd natural numbers less than 10. A = {x : x is odd natural number less than 10}
- 3. Symbols Meaning { } enclose elements in set ∈ belongs to ⊆ is a subset of (includes equal sets) ⊂ is a proper subset of ⊄ is not a subset of ⊃ is a superset of
- 4. Empty set: If a set doesn't contain any elements it is called the empty set or the null set. It is denoted by ∅ or { }. Singleton set: It is a set which contains only one element. e.g. A = {0} Finite set: It is a set which contains finite number of different elements. e.g. A = {a,e,i,o,u} Infinite set: It is a set which contains infinite number of different elements. e.g. A = {x : x ∈ set of natural numbers} Equal sets: If two or more sets contain the same elements, they are called equal sets irrespective of the order. e.g. If A = {1,2,3} and B = {2,3,1} Then A = B
- 5. Number of Elements in Set Possible Subsets Total Number of Possible Subsets {a} {a} ; ∅2 {a , b} {a , b} ; {a} , {b} , ∅4 {a , b , c} {a , b , c} , {a , b} , {a , c} , {b , c} , {a} , {b} , {c} , ∅ 8 {a , b , c, d} {a , b , c , d} , {a , b, c} , {a , b , d} , {a , c , d} , {b , c , d} , {a , b} , {a , c} , {a , d} , {a , b} …… {D} , ∅ The number of possible subsets of a set of size n is 2n 16
- 6. The Power Set (P) The power set is the set of all subsets that can be created from a given set. Example: A = {a, b, c} P (A) = {{a, b}, {a, c}, {b, c}, {a}, {b}, {c}, A, φ} Cardinal Number of A Set It is the number of elements in a set. Example: A = {a, b, c, d} n(A) = 4
- 7. A B VENN DIAGRAM Representation of sets by means of diagrams known as: Venn Diagrams are named after the English logician, John Venn. These diagrams consist of rectangles and closed curves usually circles. The universal set is represented usually by a rectangle and its subsets by circles.
- 8. A B A ∩ B A ∪ B A B This is the union symbol. It means the set that consists of all elements of set A and all elements of set B. This is the intersect symbol. It means the set containing all elements that are in both A and B
- 9. A B A - B A B A B B- A A B Difference Of Sets The difference of two sets A and B Is the set of elements which belongs to A but which do not belong to B. It is denoted by A – B. A – B = {x:x A and x B}∈ ∉ B – A = {x:x B and x A}∈ ∉
- 10. A BDisjoint Sets ACompliment of a set The two sets which do not have any elements in common are called disjoint sets. A ∪ B = ∅ U If U is the universal set, A is a subset, then compliment of A is A = {x:x U, x A}∈ ∉ or U - A
- 11. U = {1,2,3,4,5,6,7,8,9,10,11,12,13,14,15} A = {3,4,5,7,12,15} B = {2,3,4,6,7,10,14} A ∩ B = {3,7,4} A ∪ B = {2,3,4,5,6,7,10,12,14,15} A - B = {5,12,15} B – A = {2,6,10,14} 2 3 4 7 5 6 8 9 10 11 12 13 1415 A B 1 VENN DIAGRAM U
- 12. A B C A ∩ B ∩ C Only A Only B Only C Only A ∩ B not C Only B ∩ C not AOnly A ∩ C not
- 13. Commutative Laws: A ∩ B = A ∩ B and A ∪ B = B ∪ A Associative Laws: (A ∩ B) ∩ C = A ∩ (B ∩ C) and (A ∪ B) ∪ C = A ∪ (B ∪ C) Distributive Laws: A ∪ (B ∩ C) = (A ∪ B) ∩ (A ∪ C) and A ∩ (B ∪ C) = (A ∩ B) ∪ (A ∩C) Double Complement Law: (Ac )c = A De Morgan’s Laws: (A ∩ B)c = Ac ∪ Bc and(A ∪ B)c = Ac ∩ Bc
- 14. 100 people were surveyed. 52 people in a survey owned a cat. 36 people owned a dog. 24 did not own a dog or cat. universal set is 100 people surveyed C D Set C is the cat owners and Set D is the dog owners. The sets are NOT disjoint. Some people could own both a dog and a cat. 24 Since 24 did not own a dog or cat, there must be 76 that do own a cat or a dog. 52 + 36 = 88 so there must be 88 - 76 = 12 people that own both a dog and a cat. 1240 24 100 Example:
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