Mathematical Reasoning - New Statements from Old (Negation, Compound Statements)

Negation of a Statement: The denial of a statement is called its negation. If $p$ is a statement, its negation is denoted by $\sim p$ (read as 'not $p$'). For example, if $p$ is 'Every natural number is an integer', then $\sim p$ is 'It is false that every natural number is an integer' or 'There exists at least one natural number which is not an integer'. Visually, if statement $p$ represents a region in a Venn diagram, $\sim p$ represents everything outside that specific region.

Compound Statements: A compound statement is a statement which is made up of two or more simple statements, known as component statements. These are joined by connecting words like 'and', 'or', 'if-then', and 'if and only if'. For instance, 'The sun is a star and the earth is a planet' is a compound statement where the connective 'and' joins two distinct facts.

The Connective 'And' ($\wedge$): A compound statement with 'and' is true only if all its component statements are true. If even one component is false, the entire compound statement is false. In a set-theoretic visual, the 'And' operation corresponds to the intersection ($\cap$) of two sets, representing only the overlapping area where both conditions are satisfied simultaneously.

The Connective 'Or' ($\vee$): A compound statement with 'or' is true if at least one of the component statements is true. It is false only when both components are false. In logic, we usually use the 'inclusive or' (either one or both). Visually, this corresponds to the union ($\cup$) of two sets, covering the entire area occupied by either or both circles in a Venn diagram.

Conditional Statements (Implies): A statement of the form 'If $p$, then $q$' is called a conditional statement, denoted by $p \Rightarrow q$. Here, $p$ is the hypothesis and $q$ is the conclusion. This can be visualized as a directed flow or an arrow pointing from the cause to the effect. The statement is false only in the specific case where $p$ is true but $q$ is false.

Bi-conditional Statements (If and only if): A statement of the form '$p$ if and only if $q$' is denoted by $p \Leftrightarrow q$. It means both $p \Rightarrow q$ and $q \Rightarrow p$ are true. Visually, this represents a perfect equivalence or a state where two sets or conditions are exactly the same and coincide perfectly with each other.

Quantifiers: These are phrases like 'There exists' (existential quantifier, denoted by $\exists$) and 'For all' or 'For every' (universal quantifier, denoted by $\forall$). Negating a statement with 'For all' results in a statement containing 'There exists', and vice versa. For example, the negation of 'All squares are rectangles' is 'There exists a square that is not a rectangle'.

Contradiction and Tautology: 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. On a truth table, a Tautology would show a column of all 'T's, while a Contradiction would show all 'F's.