[size=150]1.[b] Proving Non-Conditional Statements:[/b][br]When proving non-conditional statements, such as "if and only if" proofs, equivalent statements, or existence proofs, we aim to establish the truth or validity of the statement without relying on conditional assumptions.[br][br]If and Only If (Iff) Proof: An "if and only if" proof establishes the equivalence of two statements by proving both the forward and backward implications. It involves proving that Statement A implies Statement B, and Statement B implies Statement A.[br]For example, to prove that two sets A and B are equal (A = B), you would prove A ⊆ B and B ⊆ A.[br][br]Equivalent Statements: Proving that two statements are equivalent means showing that they have the same truth value. It involves proving that Statement A implies Statement B, and Statement B implies Statement A, without assuming any conditional relationship between them.[br]For example, to prove that the statement "n is an even integer" is equivalent to "n[sup]2[/sup] is an even integer," you would prove that an even n implies an even n[sup]2[/sup], and an odd n implies an odd n[sup]2[/sup].[br][br]Existence Proof: An existence proof demonstrates the existence of an object satisfying a particular condition or property. It does not necessarily provide a way to construct the object but proves that at least one such object exists. [br]For example, to prove that there exists a real number x such that x[sup]2[/sup] = 2, you would show that at least one solution exists without explicitly finding or constructing the value of x.[br][br]2. [b]Constructive vs Non-Constructive Proof:[/b][br]Constructive Proof: A constructive proof not only establishes the existence of an object but also provides a method or construction to find or obtain the object explicitly.[br]For example, a constructive proof of the existence of a square root of 2 would involve providing a method to find or approximate the square root, such as using the Babylonian method or continued fractions.[br][br]Non-Constructive Proof: A non-constructive proof establishes the existence of an object without providing an explicit method or construction to find it. It relies on indirect reasoning or contradiction to demonstrate that the object must exist.[br]For example, a non-constructive proof of the existence of an irrational number between any two distinct rational numbers involves assuming the contrary (that no such irrational number exists) and arriving at a contradiction.[br][br]In both constructive and non-constructive proofs, the goal is to establish the truth or validity of a statement. However, constructive proofs provide additional information by explicitly demonstrating how to obtain or construct the desired object, while non-constructive proofs rely on logical reasoning to establish existence without necessarily providing a method of construction.[br][br]Overall, the choice between constructive and non-constructive proofs depends on the nature of the problem and the information needed to establish the statement's validity.[/size]