Mathematical Reasoning - Statements and Logical Connectives

A Mathematical Statement is a declarative sentence which is either definitely true or definitely false, but not both. For example, 'The square of 4 is 16' is a statement, while 'Give me that book' is a command and not a mathematical statement. Visually, imagine a binary switch that can only be in the 'ON' (True) or 'OFF' (False) position, never in between.

Negation of a statement $p$ is denoted by $\sim p$. If $p$ is true, $\sim p$ is false; if $p$ is false, $\sim p$ is true. This can be visualized as a simple two-column Truth Table where the input column $p$ lists values $T, F$ and the output column $\sim p$ flips them to $F, T$.

Compound Statements are formed by combining two or more simple statements using logical connectives like 'and' ($\\wedge$), 'or' ($\\vee$), 'if...then' ($\\Rightarrow$), and 'if and only if' ($\\Leftrightarrow$). Visually, this creates a branching structure or a Truth Table with $2^n$ rows, where $n$ is the number of simple statements.

Conjunction ($p \\wedge q$) is true only when both component statements $p$ and $q$ are true. In a Venn Diagram representation, this corresponds to the intersection of two sets, where the shaded area represents only the region shared by both $p$ and $q$.

Disjunction ($p \\vee q$) is false only when both component statements $p$ and $q$ are false. In all other cases, it is true. Visually, in a Venn Diagram, this is the union of two sets, covering all regions belonging to $p$, $q$, or both.

A Conditional Statement $p \\Rightarrow q$ (read as 'If $p$, then $q$') is false only in one specific case: when the antecedent $p$ is true and the consequent $q$ is false ($T \\Rightarrow F$). Visually, think of a contract: the contract is broken (False) only if the first party fulfills their duty ($p$ is $T$) but the second party does not deliver ($q$ is $F$).

Biconditional Statement $p \\Leftrightarrow q$ (read as '$p$ if and only if $q$') is true when both $p$ and $q$ have the same truth value (both True or both False). In a Truth Table, the resulting column shows 'True' for the $(T, T)$ and $(F, F)$ rows and 'False' otherwise.

Tautology and Contradiction: A compound statement that is always true regardless of the truth values of its components is a Tautology. Conversely, a statement that is always false is a Contradiction. Visually, a Tautology is a Truth Table column filled entirely with 'T', while a Contradiction is a column filled entirely with 'F'.