Set, Functions & Groups¶
Collection of well defined objects is called set.
Ways of Describing a Set¶
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Descriptive Method: In this method the set can be described in words.
For Example: The set of natural numbers.
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Tabular Method: In this method the set is described by listing its elements with in brackets.
For Example: Set of natural numbers \(=\{a,e,i,o,u\}\)
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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: