Mathematical Reasoning - Notes, Concept and All Important Formula

MATHEMATICAL REASONING

1. STATEMENT :

A declarative sentence which is either true or false but not both, is called a statement. A sentence which is an exclamatory or a wish or an imperative or an interrogative can not be a statement.

If a statement is true then its truth value is $T$ and if it is false then its truth value is $F$.

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All Chapter Notes, Concept and Important Formula

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2. SIMPLE STATEMENT:

Any statement whose truth value does not depend on other statement are called simple statement.

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3. COMPOUND STATEMENT :

A statement which is a combination of two or more simple statements are called compound statement.

Here the simple statements which form a compound statement are known as its sub statements.

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4. LOGICAL CONNECTIVES :

The words or phrases which combines simple statements to form a compound statement are called logical connectives.

$\scriptsize{ \begin{array}{|l|l|l|l|l|} \hline \text { S.N. } & \text { Connectives } & \text { Symbol } & \text { Use } & \text { Operation } \\ \hline 1 . & \text { and } & \wedge & p \wedge q & \text { conjunction } \\ 2 . & \text { or } & \vee & p \vee q & \text { disjunction } \\ 3 . & \text { not } & \sim \text { or }^{\prime} & \sim p \text { or } p ^{\prime} & \text { negation } \\ 4 . & \text { If } \ldots \text { then } \ldots . . & \Rightarrow or \rightarrow & p \Rightarrow q & \text { Implication or } \\ 5 . & \text { If and only if (iff) } & \Leftrightarrow or \leftrightarrow & \begin{array}{l} \text { or } p \rightarrow q \\ \text { conditional } \\ p \Leftrightarrow q \\ \text { or } p \leftrightarrow q \end{array} & \begin{array}{l} \text { Equivalence or } \\ \text { Bi-conditional } \end{array} \\ \hline \end{array}}$

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5. Truth Table

$\begin{array}{l} \text { Conjunction }\\ \begin{array}{|c|c|c|} \hline p & q & p \wedge q\\ \hline T & T & T \\ T & F & F \\ F & T & F \\ F & F & F \\ \hline \end{array} \end{array}$ $\begin{array}{l} \text { Disjunction }\\ \begin{array}{|c|c|c|} \hline p & q & p \vee q\\ \hline T & T & T \\ T & F & T \\ F & T & T \\ F & F & F \\ \hline \end{array} \end{array}$ $\begin{array}{l} \text {Negation }\\ \begin{array}{|c|c|} \hline p & (\sim q)\\ \hline T & F \\ F & T \\ \hline \end{array} \end{array}$ $\begin{array}{l} \text { Conditional }\\ \begin{array}{|c|c|c|} \hline p & q & p \rightarrow q\\ \hline T & T & T \\ T & F & F \\ F & T & T \\ F & F & T \\ \hline \end{array} \end{array}$ $\begin{array}{l} \text { Biconditional }\\ \begin{array}{|c|c|c|c|c|} \hline p & q & p \rightarrow q & q \rightarrow p & \begin{array}{c} ( p \rightarrow q ) \wedge( q \rightarrow p ) \\ \text { or } p \leftrightarrow q \end{array} \\ \hline T & T & T & T & T \\ T & F & F & T & F \\ F & T & T & F & F \\ F & F & T & T & T \\ \hline \end{array} \end{array}$

Note : If the compound statement contain $n$ sub statements then its truth table will contain $2^{n}$ rows.

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6. LOGICAL EQUIVALENCE :

Two compound statements $S _{1}( p , q , r \ldots)$ and $S _{2}( p , q , r \ldots .)$ are said to be logically equivalent or simply equivalent if they have same truth values for all logical possibilities.

Two statements $S _{1}$ and $S _{2}$ are equivalent if they have identical truth table i.e. the entries in the last column of their truth table are same. If statements $S _{1}$ and $S _{2}$ are equivalent then we write $S _{1} \equiv S _{2}$.

i.e. $\boxed{ p \rightarrow q \equiv \sim p \vee q}$

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7. TAUTOLOGY AND CONTRADICTION:

(i) Tautology : A statement is said to be a tautology if it is true for all logical possibilities.

i.e. its truth value always $T$. it is denoted by t.

(ii) Contradiction/Fallacy : A statement is a contradiction if it is false for all logical possibilities.

i.e. its truth value always $F$. It is denoted by $c$

Note : The negation of a tautology is a contradiction and negation of a contradiction is a tautology.

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8. DUALITY :

Two compound statements $S _{1}$ and $S _{2}$ are said to be duals of each other if one can be obtained from the other by replacing $\wedge$ by $\vee$ and $\vee$ by $\wedge$

If a compound statement contains the special variable (tautology) and c (contradiction) then we obtain its dual replacing $t$ by $c$ and $c$ by $t$ in addition to replacing $\wedge$ by $\vee$ and $\vee$ by $\wedge$.

Note:

(i) the connectives $\wedge$ and $\vee$ are also called dual of each other.

(ii) If $S^{*}( p , q )$ is the dual of the compound statement $S ( p , q )$ then

(a) $S^{*}(\sim p, \sim q) \equiv \sim S(p, q)$

(b) $\sim S^{*}(p, q) \equiv S(\sim p, \sim q)$

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9. CONVERSE, INVERSE AND CONTRAPOSITIVE OF THE CONDITIONAL STATEMENT (p $\rightarrow$ q):

(i) Converse : The converse of the conditional statement $p \rightarrow q$ is defined as $q \rightarrow p$

(ii) Inverse : The inverse of the conditional statement $p \rightarrow q$ is defined as $\sim p \rightarrow \sim q$

(iii) Contrapositive : The contrapositive of conditional statement $p \rightarrow q$ is defined as $\sim q \rightarrow \sim p$

Note $:(p \rightarrow q) \equiv(\sim q \rightarrow \sim p) \equiv(\sim p \vee q)$

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10. NEGATION OF COMPOUND STATEMENTS :

If $p$ and $q$ are two statements then

(i) Negation of conjunction : $\sim(p \wedge q) \equiv \sim p \vee \sim q$

(ii) Negation of disjunction : $\sim(p \vee q) \equiv \sim p \wedge \sim q$

(iii) Negation of conditional : $\sim(p \rightarrow q) \equiv p \wedge \sim q$

(iv) Negation of biconditional : $\sim(p \leftrightarrow q) \equiv(p \wedge \sim q) \vee(q \wedge \sim p)$

$\equiv(\sim p \leftrightarrow q ) \equiv( p \leftrightarrow \sim q )$

As we know that $p \leftrightarrow q \equiv( p \rightarrow q ) \wedge( q \rightarrow p )$

$\begin{aligned}\therefore \sim( p\leftrightarrow q ) \equiv &\sim[( p \rightarrow q ) \wedge( q \rightarrow p )]\\& \equiv \sim( p \rightarrow q ) \vee \sim( q \rightarrow p ) \\& \equiv( p \wedge \sim q ) \vee( q \wedge \sim p ) \end{aligned}$

Note : The above result also can be proved by preparing truth table for $\sim(p \leftrightarrow q)$ and $(p \wedge \sim q) \vee(q \wedge \sim p)$

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11. ALGEBRA OF STATEMENTS:

If $p , q , r$ are any three statements then the some low of algebra of statements are as follow

(i) Idempotent Laws:

(a) $p \wedge p \equiv p$

(b) $p \vee p \equiv p$

(ii) Commutative laws:

(a) $\quad p \wedge q \equiv q \wedge p$

(b) $\quad p \vee q \equiv q \vee p$

(iii) Associative laws:

(a) $\quad(p \wedge q) \wedge r \equiv p \wedge(q \wedge r)$

(b) $\quad( p \vee q ) \vee r \equiv p \vee( q \vee r )$

(iv) Distributive laws:

(a) $\quad p \wedge( q \vee r ) \equiv( p \wedge q ) \vee( p \wedge r )$

(b) $\quad p \vee( q \wedge r ) \equiv( p \vee q ) \wedge( p \vee r )$

(v) De Morgan Laws:

(a) $\sim(p \wedge q) \equiv \sim p \vee \sim q$

(b) $\quad \sim(p \vee q) \equiv \sim p \wedge \sim q$

(vi) Involution laws (or Double negation laws) : $\sim(\sim p ) \equiv p$

(vii) Identity Laws : If $p$ is a statement and $t$ and $c$ are tautology and contradiction respectively then

(a) $p \wedge t \equiv p$

(b) $p \vee t \equiv t$

(c) $p \wedge c \equiv c$

(d) $p \vee c \equiv p$

(viii) Complement Laws:

(a) $p \wedge(\sim p ) \equiv c$

(b) $p \vee(\sim p ) \equiv t$

(c) $(\sim t ) \equiv c$

(d) $(\sim c) \equiv t$

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12. QUANTIFIED STATEMENTS AND QUANTIFIERS :

The words or phrases "All", "Some", "None", "There exists a" are ex-amples of quantifiers. 

A statement containing one or more of these words (or phrases) is a quantified statement.

Note : Phrases "There exists a" and "Atleast one" and the word "some"

have the same meaning.

NEGATION OF QUANTIFIED STATEMENTS :

(1) 'None' is the negation of 'at least one' or 'some' or 'few' Similarly negation of 'some' is 'none'

(2) The negation of "some $A$ are $B ^{\prime \prime}$ or "There exist $A$ which is $B ^{\prime \prime}$ is "No $A$ are (is) $B ^{\prime \prime}$ or "There does not exist any A which is $B ^{\prime \prime}$.

(3) Negation of "All A are B" is "Some A are not B".

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