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- https://math.libretexts.org/Bookshelves/Mathematical_Logic_and_Proof/Book%3A_Mathematical_Reasoning__Writing_and_Proof_(Sundstrom)/03%3A_Constructing_and_Writing_Proofs_in_Mathematics/3.01%3A_Direct_ProofsIf a statement is true, then write a formal proof of that statement, and if it is false, then provide a counterexample that shows it is false. (a) For each integer a, if there exists an integer \(...If a statement is true, then write a formal proof of that statement, and if it is false, then provide a counterexample that shows it is false. (a) For each integer a, if there exists an integer n such that a divides (8n + 7) and a divides (4n + 1), then a divides 5.
- https://math.libretexts.org/Courses/SUNY_Schenectady_County_Community_College/Discrete_Structures/03%3A_Constructing_and_Writing_Proofs_in_Mathematics/3.01%3A_Direct_ProofsIf a statement is true, then write a formal proof of that statement, and if it is false, then provide a counterexample that shows it is false. (a) For each integer a, if there exists an integer \(...If a statement is true, then write a formal proof of that statement, and if it is false, then provide a counterexample that shows it is false. (a) For each integer a, if there exists an integer n such that a divides (8n + 7) and a divides (4n + 1), then a divides 5.
- https://math.libretexts.org/Courses/Monroe_Community_College/MTH_220_Discrete_Math/3%3A_Proof_Techniques/3.2%3A_Direct_ProofsThe notation a | b represents a relationship between the integers a and b and is simply a shorthand for “a divides b.” "Divides" as in a | b is a relation (true or false), whil...The notation a | b represents a relationship between the integers a and b and is simply a shorthand for “a divides b.” "Divides" as in a | b is a relation (true or false), while "divided by" as in \dfrac{a}{b} or a/b is an operation (results in a number) .
- https://math.libretexts.org/Bookshelves/Combinatorics_and_Discrete_Mathematics/A_Spiral_Workbook_for_Discrete_Mathematics_(Kwong)/03%3A_Proof_Techniques/3.02%3A_Direct_ProofsThe first one is the fallacy of the inverse or the denial of the antecedent: \begin{array}{cl} & p \Rightarrow q \\ & \overline{p} \\ \hline \therefore & \overline{q} \end{array} This in effect pr...The first one is the fallacy of the inverse or the denial of the antecedent: \begin{array}{cl} & p \Rightarrow q \\ & \overline{p} \\ \hline \therefore & \overline{q} \end{array} This in effect proves the inverse \overline{p}\Rightarrow \overline{q}, which we know is not logically equivalent to the original implication.
