Deductive Logic and Proving Mathematical Statements

The question asks about the type of logic used when we prove that the sum of two even numbers is an even number, specifically by using known results, definitions, and rules of inference.

Let's understand the two types of logic mentioned:

• Inductive Logic: This type of logic involves reasoning from specific instances to reach a general conclusion. The conclusion is likely, but not guaranteed, to be true based on the observed patterns. For example, observing that 2+4=6 (even), 6+10=16 (even), and 12+8=20 (even) might lead you to *induce* that the sum of two even numbers is always even. However, this is not a formal proof.
• Deductive Logic: This type of logic involves reasoning from one or more general statements (premises) that are known or assumed to be true, to reach a specific conclusion that logically follows from those premises. If the premises are true and the reasoning is valid, the conclusion must be true. Mathematical proofs typically rely heavily on deductive logic.

When we formally prove a statement like "the sum of two even numbers is an even number" using known results, definitions, and rules of inference, we are starting with established truths (definitions of even numbers, rules of arithmetic) and using logical steps (rules of inference) to arrive at the conclusion. This process guarantees the truth of the conclusion, provided the starting points and steps are valid.

For example, the definition of an even number is an integer \( n \) that can be written as \( n = 2k \) for some integer \( k \). If we take two even numbers, say \( a \) and \( b \), then by definition, \( a = 2k_1 \) and \( b = 2k_2 \) for some integers \( k_1 \) and \( k_2 \). Their sum is \( a + b = 2k_1 + 2k_2 = 2(k_1 + k_2) \). Since \( k_1 + k_2 \) is also an integer (by rules of arithmetic), the sum \( a + b \) fits the definition of an even number. This step-by-step reasoning using definitions and arithmetic rules is a clear example of deductive logic.

Since the proof method described (using known results, definitions, and rules of inference) aligns perfectly with the process of deductive reasoning, the logic being used is Deductive Logic.

Therefore, only Deductive Logic is used in this method of proof.