Mathematical Preliminaries

Tejas N SAssistant Professor in Mathematics, NIE First Grade College, MysuruEmail: nstejas@gmail.com Mobile: +91 9845410469

• Definition: A simple statement which clearly explains the meaning. Definition must be clear and very precise.

• Theorem: A mathematical statement that is proved using mathematical reasoning.

• Lemma: A minor result whose sole purpose is to help in proving a theorem.

• Corollary: A result in which the (usually short) proof relies on a given theorem.

• Proposition: A proved and often interesting result, but generally less important than a theorem.

• Conjecture: A statement that is unproved, but is believed to be true.

• Claim: An assertion that is then proved. It is often used like an informal lemma.

• Axiom/Postulate: A statement that is assumed to be true without proof.

• To know whether a theorem is true or false, we have to see whether the theorem follows or does not follow from the set of axiom of choices which are assumed to be true.

• Methods of proofs

• Direct proofs: We start with a set of axioms or definitions or previously established result or construct a logical sequence of statements to prove the result.

• Indirect proofs: We assume that the given statement is false (i.e., negation of the given statement) and then by logical sequence of statements we will arrive at some contradictions.

• If statement P is true and if statement P implies statement Q then statement Q is true. (Rule of detachment)

• If statement P implies statement Q and Q implies statement R then P implies R. (Rule of syllogism) i.e., P $\Rightarrow$ Q and Q $\Rightarrow$ R then P $\Rightarrow$ R

• If statement P implies statement Q and Q implies P then P if and only (iff) if Q. i.e., P $\Rightarrow$ Q and Q $\Rightarrow$ P then P $\Leftrightarrow$ Q

• If statement P implies statement Q is true iff negation Q implies negation P. i.e., P $\Rightarrow$ Q $\Leftrightarrow$ $\sim$Q $\Rightarrow$ $\sim$P

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• Georg Cantor, the founder of set theory gave the definition of a set as "A set is a gathering together into a whole of definite distinct objects of our perception or of our own thoughts - which are called the elements of the set"

• A set is a collection of well - defined objects.

• In practice, we use capital letters to denote the set and small letters to denote the elements of set.

• Ways of describing a set

• By listing its elemets: We write all the elements of the set without repetition and enclose within a pair of flower brackets. We can write the elements in any order.
Example 1: $S=\{1,2,3,4,5,6,7,8,9\}$
Example 2: $A=\{53,35,-6,0,2,1,-10\}$

• By rule method: We describe the properties of the elements of the set.
Example: $A=\{x: x$ is a positive integer less than 10$\}$

• By recursion: We define the elements of the set by a computational rule for calculating the elements.
Example: $A=\{a_n: a_0=1, a_{n}=a_{n-1}\}$

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• If an element $a$ is an element of a set $A$ then we symbolically write as $a \in A$, otherwise we write $a\notin A$

• If a set has finite number of elements then the it is called as finite set.

• If a set has infinite number of elements then the it is called as infinite set.

• A set with no element is called an empty/null set denoted by $\{\}$ or $\phi$

• A set with single element is called singleton set.

• A set $A$ is said to be a subset of set $B$ if every element of $A$ is also an element of $B$ denoted by $A\subseteq B$. A set $A$ is not a subset of set $B$ if $\exists$ atleast one element in $A$ which is not in $B$ denoted by $A\nsubseteq B$.
Example 1: Let $A=\{1,2,3,4,5\}$ and $B=\{0,1,2,3,4,5,6\}$ then $A\subseteq B$
Example 2: Let $A=\{a,b,c,d,e,f\}$ and $B=\{a,b,c\}$ then $B\subseteq A$
Example 3: Let $X=\{1,2,3\}$ and $Y=\{2,3\}$ then $X\nsubseteq Y$ but $Y\subseteq X$
Example 4: Let $S_1=\{1,2,3,4\}$ and $S_2=\{5,6,7,8\}$ then $S_1\nsubseteq S_2$ and $S_2\nsubseteq S_1$
Remark: A set is always a subset of itself and empty set is a subset of every set (trivial subsets). So every set has atleast two subsets.

• A set $A$ is said to be a proper subset of set $B$ if $B$ has atleast one element more than that of $A$ denoted by $A\subset B$.
Example: Let $A=\{1,2,3,4\}$ and $B=\{1,2\}$ then $B\subset A$

• Let $A$ be a set, then power set of $A$ denoted by $P(A)$ is given by $\{S:S\subseteq A\}$
Example: Let $A=\{1,2,3\}$ then $S_1=\{1,2,3\}$, $S_2=\{\}$, $S_3=\{1\}$, $S_4=\{2\}$, $S_5=\{3\}$, $S_6=\{1,2\}$, $S_7=\{1,3\}$, $S_8=\{2,3\}$ are subsets of $A$. $P(A)=\{S_1,S_2,S_3,S_4,S_5,S_6,S_7,S_8\}$

• The number of elements of a set $A$ is called the cardinality of a set $A$ denoted by $|A|$
Example: Let $A=\{a,b,c,d\}$ then $|A|=4$

• Two sets $A$ and $B$ are equal if both the sets have exactly the same elements.
In practice, to prove $A=B$ we show that $A\subseteq B$ and $B\subseteq A$

• Two sets $A$ and $B$ are equalivalent if both the sets have same cardinality. Note that equivalent sets may not be equal.

• Operations on sets

• $A\cup B=\{x: x \in A$ or $x \in B \}$ is called union of $A$ and $B$

• $A\cap B=\{x: x \in A$ and $x \in B \}$ is called intersection of $A$ and $B$

• $A-B=\{x: x \in A$ and $x \notin B \}$ is called complement of $B$ in $A$

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• Two sets are said to be disjoint if there is no common element in $A$ and $B$. In practice to prove two sets $A$ and $B$ are disjoint we show that $A\cap B=\phi$

• Standard sets of numbers

• Set of natural numbers $\mathbb{N}=\{1,2,3,4,...\}$

• Set of whole numbers $\mathbb{W}=\{0,1,2,3,4,...\}=\{0\} \cup \mathbb{N}$

• Set of integers $\mathbb{Z}=\{0,\pm 1,\pm 2,\pm 3,\pm 4,...\}$

• Set of positive integers $\mathbb{Z}^+=\{1,2,3,4,...\}$

• Set of negative integers $\mathbb{Z}^-=\{-1,-2,-3,-4,...\}$

• Set of even integers $2\mathbb{Z}=\{0,\pm 2,\pm 4,\pm 6,....\}$

• Set of rationals $\mathbb{Q}=\big\{\frac{p}{q}: p,q \in \mathbb{Z}$ and $q\neq 0\big\}$

• Set of irrationals $\mathbb{I}=\{x: x\notin \mathbb{Q}\}$

• Set of real numbers $\mathbb{R}=\mathbb{Q}\cup \mathbb{I}$

• Set of complex numbers $\mathbb{C}=\{a+ib: a,b \in \mathbb{R}$ and $i=\sqrt{-1}\}$

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• Remark 1: Every real number is a complex number but converse may not be true.

• Remark 2: $\mathbb{N}\subseteq \mathbb{W}\subseteq \mathbb{Z}\subseteq \mathbb{Q}\subseteq \mathbb{R}\subseteq \mathbb{C}$, $\mathbb{Q}\nsubseteq \mathbb{I}$, $\mathbb{I}\subseteq \mathbb{R}$.

• A function/mapping $f$ from a set $X$ into set $Y$ is a rule which assigns every element $x$ in $X$ into a unique element $y$ in $Y$ denoted by $f(x)=y$. The element $f(x)$ is called the \textbf{image} of $x$ under the mapping $f$. Mapping is denoted by $f:X\rightarrow Y$

• If $f:X\rightarrow Y$ is a mapping then $X$ is called domain of $f$ and $Y$ is called co-domain of $f$.

• Mapping $f:X \rightarrow Y$ is well-defined if all the elements of $X$ are mapped to some elements of $Y$ and an element of $X$ should be mapped to a unique element of $Y$.
In practice, to prove that the mapping $f:X\rightarrow Y$ is well-defined we show that $x_1=x_2 \Rightarrow f(x_1)=f(x_2)$

• Mappings can be defined either by giving the images of all elements of $X$ or by a computational rule which computes $f(x)$ once the $x$ is given.
Example 1: $f:[1,2,3,4]\rightarrow [a,b,c,d]$ defined by $f(1)=d$, $f(2)=a$, $f(3)=a$, $f(4)=b$
Example 2: $f:\mathbb{Z}^+\rightarrow \mathbb{R}$ defined by $f(x)=x^2-x+1$

• A mapping $f:X\rightarrow Y$ is said to be one to one mapping if different elements of $X$ are mapped into different elements of $Y$.

• A mapping $f:X\rightarrow Y$ is said to be one to many mapping if two or more elements of $X$ are mapped to same element in $Y$.

• A mapping $f:X\rightarrow Y$ is said to be into mapping if $\exists$ one element in $Y$ which is not a $f$ image of any element of $X$.

• A mapping $f:X\rightarrow Y$ is said to be onto mapping if every element in $Y$ is the $f$ image of atleast one element in $X$.

• A mapping $f:X\rightarrow Y$ is said to be identity mapping if $f(x)=x$ $\forall$ $x \in X$.

• If $f:X\rightarrow Y$ is one-one and onto mapping then $f^{-1}$ exists called the inverse mapping of $f$ defined from $Y$ to $X$. i.e., $f^{-1}:Y\rightarrow X$

• Let $X$ and $Y$ be any two sets. Let $x\in X$ and $y \in Y$, then $(x,y)$ denotes an ordered pair whose first component is $x$ and second component is $y$. Note that $(x,y)$ is different from $(y,x)$.

• Let $X$ and $Y$ be any two non empty sets, then the set of all ordered pairs $(x,y)$ is called the cartesian product of $X$ and $Y$ denoted by $X\times Y$.
i.e., $X\times Y=\{(x,y):x\in X$ and $y \in Y\}$

• A relation $R$ in a set $S$ is a collection of ordered pair of elements of $S$ (i.e., a subset of $S\times S$). When $(x,y)$ is in $R$ we write $xRy$.
Example: A relation $R$ in $\mathbb{Z}$ can be defined by $xRy$ if $x>y$.

• A relation $R$ in a set $S$ is called equivalence relation if it is
1) Reflexive: $xRx$ $\forall$ $x \in S$
2) Symmetric: $xRy$ $\Rightarrow$ $yRx$ $\forall$ $x,y \in S$
3) Transitive: $xRy$ and $yRz$ $\Rightarrow$ $xRz$ $\forall$ $x,y,z \in S$

• A binary operation $*$ on a set $S$ is a rule (function) which assigns every ordered pair $(a,b)$ of $S$, a unique element denoted by $a*b$.

• Closure law: If $*$ is a binary operation defined on set $S$ and $a*b \in S$ then $*$ is said to be closed and we say that $S$ is closed w.r.t binary operation $*$.

• Associative law: If $*$ is a binary operation defined on set $S$ then $a*(b*c)=(a*b)*c$

• Existence of identity element (Identity law): If $*$ is a binary operation defined on set $S$ and if $\exists$ an element $e \in S$ such that $a*e=e*a=a$ $\forall$ $a \in S$. $e$ is called the identity element of $S$.

• Existence of inverse element (Inverse law): If $*$ is a binary operation defined on set $S$ and for each $a\in S$ $\exists$ $a^{-1} \in S$ such that $a*a^{-1}=a^{-1}*a=e$. $a^{-1}$ is called the inverse element of $a$.

• Commutative law: If $*$ is a binary operation defined on set $S$ then $a*b=b*a$.

• An non zero integer $y$ is said to be a divisor of integer $x$ if $\exists$ an integer $z$ such taht $x=yz$. Then we write $y|x$ (read as $y$ divides $x$).

• An positive integer $p$ is said to be prime if its only divisors are 1 and $p$.