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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 \(F\).


2. SIMPLE STATEMENT:

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


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.


4. LOGICAL CONNECTIVES :

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

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


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.


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. \(\boxed{ p \rightarrow q \equiv \sim p \vee q}\)


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.


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


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


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 \((p \wedge \sim q) \vee(q \wedge \sim p)\)


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


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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