Elementary Mathematics: Set Theory & Real Numbers PDF

ELEMENTARY MATHEMATICS Set theory and real number system  Subset  Definition: set whose elements are within another given set  Example: B is the subset of A if B={1,2} and A={1,2,3,4}  Associated formulae: To find the total number of subsets, we use 2n-1,...

ELEMENTARY MATHEMATICS Set theory and real number system  Subset  Definition: set whose elements are within another given set  Example: B is the subset of A if B={1,2} and A={1,2,3,4}  Associated formulae: To find the total number of subsets, we use 2n-1, assuming ‘n’ is the number of elements in the given set.  Venn diagram  Definition: diagram representing sets by circles  Example:  Types of set  Empty set  Finite set  Infinite set  Overlapping set  Disjoint set  Universal set  Union of sets  Intersection of sets  Complement of a set  Difference of sets  Empty set  Definition: unique set that contains no elements  Example: A={composite numbers between 1 and 4}  Notation: A={}, A={Φ}  Finite set  Definition: A set with only finitely many members  Example: A={first two natural numbers}  Infinite set  Definition: a set whose elements can not be counted  Example: A={a set of natural numbers}  Overlapping set  Definition: sets that have at least one element in common  Example: A={2,3,5,7} and B={1,2,3,4}  Venn diagram: A B 5 2 1 7 3 4  Disjoint set  Definition: A family of sets sharing no elements in common; sets whose intersection is the empty set.  Example: A={2,3,5,7} and B={4,6,8,9}  Venn diagram: A B 2 4 3 6 5 8 7 9  Universal set  Definition: A set large enough to contain all sets under consideration in the current context.  Example: In the following Venn diagram, U={2,3,5,7,4,6,8,9} A B 2 4 3 6 5 8 7 9  Union of sets  Definition: The union of two sets X and Y is equal to the set of elements that are present in set X, in set Y, or in both the sets X and Y.  Example: if X={1,2} and Y={3,4}, XUY={1,2,3,4}  Intersection of sets  Definition: the set of all those elements which are common to both A and B  Example: if A={2,3,5,7} and B={1,2,3,4,5}, A ∩ B={2,3,5}  Complement of a set  Definition: the set of elements that are not in the original set  Example: if U={1,2,3,4,5} and A={2,3,5}, Ac={1,4}  Difference of sets  Definition: the set of elements that are in A but not in B  Example: if U={1,2,3,4,5} and A={2,3,5}, U-A={1,4}  Cardinality of a set  Definition: The number of elements a given set contains.  Example: if A={2,3,5}, n(A)=3