Proof Templates 1 Direct proof of an if-then theorem.19 2 Direct proof of an if-and-only-if theorem.23 3 Refuting a false if-then statement via a counterexample. 26 4 Truth table proof of logical equivalence.30 5 Proving two sets are equal.51 6 Proving one set is a subset of another. 54 7 Proving existential statements.59 8 Proving universal statements. 60 9 Combinatorial proof.67 10 Using inclusion-exclusion. 129 11 Proof by contrapositive.136 12 Proof by contradiction.137 13 Proving that a set is empty.140 14 Proving uniqueness.140 15 Proof by smallest counterexample. 146 16 Proof by the Well-Ordering Principle. 150 17 Proof by induction.158 18 Proof by strong induction. 163 19 To show f:A→B. 196 20 Proving a function is one-to-one. 199 21 Proving a function is onto.201 22 Proving two functions are equal. 213 23 Proving (G,*is a group. 344 24 Proving a subset of a group is a subgroup.354 25 Proving theorems about trees by leaf deletion. 418Proof Templates 1 Direct proof of an if-then theorem. 19 2 Direct proof of an if-and-only-if theorem. 23 3 Refuting a false if-then statement via a counterexample. 26 4 Truth table proof of logical equivalence. 30 5 Proving two sets are equal. 51 6 Proving one set is a subset of another. 54 7 Proving existential statements. 59 8 Proving universal statements. 60 9 Combinatorial proof. 67 10 Using inclusion-exclusion. 129 11 Proof by contrapositive. 136 12 Proof by contradiction. 137 13 Proving that a set is empty. 140 14 Proving uniqueness. 140 15 Proof by smallest counterexample. 146 16 Proof by the Well-Ordering Principle. 150 17 Proof by induction. 158 18 Proof by strong induction. 163 19 To show f : A ~ B. 196 20 Proving a function is one-to-one. 199 21 Proving a function is onto. 201 22 Proving two functions are equal. 213 23 Proving (G, *) is a group. 344 24 Proving a subset of a group is a subgroup. 354 25 Proving theorems about trees by leaf deletion. 418