Set and Logic

Set and notation

A well defined collection of objects is known as a set. The set is denoted by the capital letters and its elements by small letters.

Subset and proper subset

A set A is subset of set B, if every elements of set A is also an element of B. It is represented by A Í B.

A set A is proper subset of set B if A is subset of B and A ¹ B. It is denoted by A Ì B and set A is said to proper subset of B and set B is said to be super set of A.

Universal set

A fixed set such all the sets under consideration are the subsets of that fixed set, then the fixed set is called the universal set. The universal set is denoted by U.

Empty or Null set

A set having no element is called empty or null set. It is denoted by f or { }.

Finite set and infinite set

A set having a finite number of elements is known as a finite set. The set which consists infinite number of elements is called an infinite set.

Equal and Equivalent sets

Two sets A and B are said to be equal if they have the same elements. If A and B are equal then, we write A = B. Thus, if A Í B and B Í A, then A = B.

Two sets A and B are said to be equivalent if they have the same number of elements. The equivalent sets A and B are denoted by A~B.

Intersecting and disjoint sets

If two sets A and B have at least one element in common, then A and B are called intersecting sets. If sets A and B have no elements in common, then these sets are called disjoint sets.

Power set

The collection of all possible subsets of any set S is called the power set of S. The power set of S is denoted by 2^s. If S = {a, b}, then 2^S = {f, {a}, {b}, {a, b}}.

Venn diagram

The diagrammatic representation of sets, relation of sets and operation on sets is known as Venn diagram. It consists of a universal set U represented by a rectangle, subset of U by the closed curve and the elements of set by the points within the closed curve.

Operation on sets:

· Union of two sets: The union of the sets A and B is defined as the set of all elements which belong to A or B or both.

In symbol, A È B = {x : x ÎA or x Î B}.

· Intersection of two sets: The intersection of sets A and B is defined as the set of all elements which belong to both sets A and B.

In symbol, A Ç B = {x: xÎ A and x ÎB}

· Difference of two sets: The difference of set B from set A is denoted by A - B is a set which consists all elements of set A but not belonging to set B.

In symbol, A - B = {x: x Î A and x Ï B}

Solved Problems