Mathematical Reasoning - Statements

A mathematical statement is a sentence which is either true or false, but not both. Sentences involving variables like 'He', 'She', or 'It', or relative terms like 'today' or 'here', are generally not statements unless the context fixes them. Visualize a decision tree where a sentence only qualifies as a statement if it follows a single path to either a 'True' or 'False' leaf node, without any ambiguity.

Negation of a statement involves denying the statement. If $p$ is a statement, its negation is denoted by $\sim p$ (read as 'not $p$'). If $p$ is true, $\sim p$ is false, and vice versa. This can be visualized using a two-row truth table where the value in the second column is the exact opposite of the first.

Compound statements are formed by combining two or more simple statements using logical connectives such as 'and' (conjunction, denoted by $\wedge$) and 'or' (disjunction, denoted by $\vee$). Think of 'and' as a series electrical circuit where the bulb glows only if both switches are closed, and 'or' as a parallel circuit where the bulb glows if at least one switch is closed.

Quantifiers are phrases like 'There exists' (existential quantifier, $\exists$) and 'For all' (universal quantifier, $\forall$). They define the scope of the statement. Visually, 'For all' covers every single point within a defined boundary (like a shaded circle), whereas 'There exists' highlights at least one specific point within that boundary.

Implications or Conditional statements are of the form 'If $p$, then $q$', denoted by $p \Rightarrow q$. Here, $p$ is the hypothesis and $q$ is the conclusion. This relationship is like a one-way street: the truth of $p$ forces the truth of $q$, but the truth of $q$ does not necessarily mean $p$ is true.

The Contrapositive and Converse are variations of the conditional statement $p \Rightarrow q$. The Contrapositive is $\sim q \Rightarrow \sim p$, which is logically equivalent to the original statement. The Converse is $q \Rightarrow p$, which may or may not be true. Visually, if $p \Rightarrow q$ represents 'All squares are rectangles', the contrapositive is 'If it is not a rectangle, it is not a square'.

A Bi-conditional statement is of the form '$p$ if and only if $q$', denoted by $p \Leftrightarrow q$. This means both $p \Rightarrow q$ and $q \Rightarrow p$ are true. This can be visualized as a two-way arrow, indicating that $p$ and $q$ are logically identical in terms of truth value.

Validating statements can be done through direct proof, contrapositive proof, or proof by contradiction. In a proof by contradiction, we assume the negation of the statement is true and work towards a logical impossibility (e.g., $1 = 0$), thereby proving the original statement must be true.