In the text the commutativity of,+ i+k=k+i, was proved by induction on k (see Theorem 3-9). Prov

step 1 This question is from mathematical induction. Here given us s n this is 3 plus 9 plus 27 plus 3n

In the text the commutativity of,+ i+k=k+i, was proved by...

Key Concepts

Commutativity of Addition

The commutativity of addition is the property that the order of the operands does not affect the sum, meaning that for all natural numbers i and k, i + k equals k + i. Establishing this property, particularly via different inductive approaches (induction on k versus induction on i), reinforces the inherent symmetry in the operation and ensures consistency within arithmetic as defined by the Peano axioms.

Inductive Hypothesis

The inductive hypothesis is the assumption that a statement or property holds for a particular, but arbitrary, case. In the context of proving commutativity of addition by induction on one of the operands, the inductive hypothesis allows you to assume that the property (e.g., i + k = k + i) holds for a given i, and then use that assumption to prove the property for the successor of i. This step is pivotal for translating the base case validity into a universal one.

Mathematical Induction

Mathematical induction is a proof technique used to establish a given statement for all natural numbers. It involves proving a base case and then showing that if the statement holds for an arbitrary natural number, it must also hold for its successor. This method is essential when dealing with recursively defined operations or sequences, as it ensures the property in question is maintained throughout the entire domain.

Recursive Definition of Addition

In Peano arithmetic, addition is defined recursively: adding zero to any number returns the number itself, and adding the successor of a number is defined by taking the successor of the sum. This framework not only provides a concise definition of addition but also enables proofs by induction, as it clearly exhibits the case for zero and for the 'next' number, aligning perfectly with the structure of inductive proofs.

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