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02 - Set, Functions & Groups

Set, Functions & Groups

Collection of well defined objects is called set.

Ways of Describing a Set

  • Descriptive Method: In this method the set can be described in words.

    For Example: The set of natural numbers.

  • Tabular Method: In this method the set is described by listing its elements with in brackets.

    For Example: Set of natural numbers \(=\{a,e,i,o,u\}\)

  • Set Builder Method: In this method a symbol or letter is used for an arbitrary member of the set and stating the property common to all members of the set.

    For Example: Set of natural numbers \(=\{x | x\) is a vowel of the English alphabets \(\}\)

Set of Numbers

  • Natural numbers \(N=\{1,2,3,...\}\)
  • Whole numbers \(W=\{0,1,2,3,...\}\)
  • Integers \(Z=\{0,\pm1,\pm2,\pm3,...\}\)
  • Positive integers \(Z^+=\{0,+1,+2,+3,...\}\)
  • Negative integers \(Z^-=\{0,-1,-2,-3,...\}\)
  • Odd numbers \(O=\{1,3,5,...\}\)
  • Even numbers \(E=\{0,2,4,...\}\)
  • Prime numbers \(E=\{2,3,5,7,11,13,...\}\)
  • Rational numbers $$ Q=\lbrace x | x=\frac{p}{q}, p, q \in Z, q\rlap{/}{=}0 \rbrace $$
  • Irrational numbers $$ Q'=\lbrace x | x\rlap{/}{=}\frac{p}{q}, p, q \in Z, q\rlap{/}{=}0 \rbrace $$
  • Real numbers \(R=Q \cup Q'\)

Terms Regarding Sets

  • Order of Set: The number of elements present in a set is called order of set.
  • Finite Set: If a set has definite number of elements present in it. Example: \(\{1,2,3,...,999\}\)
  • Infinite Sets: If a set has indefinite number of elements present in it. Example: \(\{1,2,3,...,n\}\)
  • Null or Empty Set: A set having no element. It is denoted by \(\phi\).
  • Singleton Set: A set having only one element. Example: \(\{7\}\).
  • Equal Sets: Two sets \(A\) & \(B\) are equal iff they have same elements. Example: If \(A=\{1,2,3\}\) & \(B=\{2,1,3\}\) then \(A=B\).
  • Equivalent or Similar Set: Two sets \(A\) & \(B\) are equivalent iff a one to one correspondence exits between the elements of the sets. Example: If \(A=\{1,2,3,4,5\}\) & \(B=\{a,e,i,o,u\}\) then \(A~B\).
  • Subset: If every element of a set \(A\) is an element of set \(B\), then \(A\) is a subset of \(B\) written as \(A\subseteq B\). Example: If \(A=\{1,2,3\}\) & \(B=\{1,2,3,...,10\}\) then \(A\subseteq B\).
  • Proper Subset: If \(A\subseteq B\) and \(B\) contains at least one element which is not an element of set \(A\) then \(A\) is called proper subset of \(B\). Example: If \(A=\{1,2,3\}\) & \(B=\{1,2,3,...,10\}\) then \(A\subseteq B\).
  • Improper Subset: Two equal set are called Improper Subsets.
  • Power Set: The collection of all proper and improper subsets of a set is called power set. It is denoted by \(P(A)\). If a set has \(n\) elements its power set will has \(2^n\) elements. Example: If \(A=\{a,b\}\) then \(P(A)=\)\(\{\phi, \{a\}, \{b\}, \{a,b\} \}\)
  • Universal Set or Universe of Discourse: The super set of all the sets under a discussion. It is denoted by \(U\).

Info

Natural numbers are subset of Integers, are subset of Rational numbers, are subset of Real numbers, are subset of Complex numbers $$ N\subset Z\subset Q\subset R \subset C $$

Operations on Sets

  • Union of Sets: Union of sets \(A\) & \(B\), denoted by \(A\cup B\), is the set of all the elements belongs to both \(A\) & \(B\). Symbolically: $$ A\cup B= \{x | x \in A \land x\in B \} $$
  • Intersection of Sets:

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