Class 11: Mathematical Reasoning

$\displaystyle \textbf{1. Statement}$
$\displaystyle \text{A sentence is called a mathematically acceptable statement or simply a statement if it is}$
$\displaystyle \text{either true or false but not both.}$
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$\displaystyle \textbf{2. Negation of a Statement}$
$\displaystyle \text{The denial of a statement is called negation of the statement. The negation of a statement }p$
$\displaystyle \text{is denoted by }\sim p\text{ and is read as "not }p\text{".}$
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$\displaystyle \textbf{3. Compound Statement}$
$\displaystyle \text{A statement is called a compound statement if it is made up of two or more simple}$
$\displaystyle \text{statements. The simple statements are called component statements of the compound}$
$\displaystyle \text{statement.}$
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$\displaystyle \textbf{4. Formation of Compound Statements}$
$\displaystyle \text{Compound statements are obtained by using connecting words like "and", "or" etc. and}$
$\displaystyle \text{phrases "If-then", "Only if", "if and only if", "There exists", "For all" etc.}$
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$\displaystyle \textbf{5. Compound Statement with "AND"}$
$\displaystyle \text{(i) true if all its component statements are true.}$
$\displaystyle \text{(ii) false if any of its component statements is false.}$
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$\displaystyle \textbf{6. Compound Statement with "OR"}$
$\displaystyle \text{(i) true when one component statement is true or both the component statements are true.}$
$\displaystyle \text{(ii) false when both the component statements are false.}$
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$\displaystyle \textbf{7. Equivalent Forms of "If }p\text{, then }q\text{"}$
$\displaystyle \text{A sentence with "If }p\text{, then }q\text{" can be written in the following ways:}$
$\displaystyle \text{(i) }p\text{ implies }q\text{ (denoted by }p\Rightarrow q\text{)}$
$\displaystyle \text{(ii) }p\text{ is sufficient condition for }q$
$\displaystyle \text{(iii) }q\text{ is necessary condition for }p$
$\displaystyle \text{(iv) }p\text{ only if }q$
$\displaystyle \text{(v) }\sim q\text{ implies }\sim p$
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$\displaystyle \textbf{8. Contrapositive, Converse and Inverse}$
$\displaystyle \text{(i) The contrapositive of the statement }p\Rightarrow q\text{ is the statement }\sim q\Rightarrow\sim p.$
$\displaystyle \text{(ii) The converse of the statement }p\Rightarrow q\text{ is the statement }q\Rightarrow p.$
$\displaystyle \text{(iii) The inverse of the statement }p\Rightarrow q\text{ is the statement }\sim p\Rightarrow\sim q.$
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$\displaystyle \textbf{9. Quantifiers}$
$\displaystyle \text{For all or for every is called universal quantifier. There exists is called existential quantifier.}$
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$\displaystyle \textbf{10. Valid and Invalid Statements}$
$\displaystyle \text{A statement is said to valid or invalid according as it is true or false.}$
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$\displaystyle \textbf{11. Truth Conditions for Statements}$
$\displaystyle \text{If }p\text{ and }q\text{ are two mathematical statements, then the statement}$
$\displaystyle \text{(i) "}p\text{ and }q\text{" is true if both }p\text{ and }q\text{ are true.}$
$\displaystyle \text{(ii) "}p\text{ or }q\text{" is true if }p\text{ is false }\Rightarrow q\text{ is true or, }q\text{ is false }\Rightarrow p\text{ is true.}$
$\displaystyle \text{(iii) "If }p\text{, then }q\text{" is true if}$
$\displaystyle \text{(a) }p\text{ is true }\Rightarrow q\text{ is true}$
$\displaystyle \text{or}$
$\displaystyle \text{(b) }q\text{ is false }\Rightarrow p\text{ is false}$
$\displaystyle \text{or}$
$\displaystyle \text{(c) }p\text{ is true and }q\text{ is false leads us to a contradiction.}$
$\displaystyle \text{(iv) "}p\text{ if and only if }q\text{" is true, if}$
$\displaystyle \text{(a) }p\text{ is true }\Rightarrow q\text{ is true}$
$\displaystyle \text{and}$
$\displaystyle \text{(b) }q\text{ is true }\Rightarrow p\text{ is true.}$
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