Question

Let S denote a nonempty set, and let N(S) denote the set of one-to-one mappings from S to S. Which theorems prove that composition is an associative operation on N(S)?

Answer 1

The Associative Property of Composition proves that composition is an associative operation on N(S) in mathematics.

Step-by-step explanation:

The theorem that proves composition is an associative operation on N(S) is called the Associative Property of Composition. This property states that if we have three functions f, g, and h in N(S), then the composition (f ° g) ° h is equal to f ° (g ° h). In other words, the order of composition does not matter.

For example, let's say we have the functions f: S → S, g: S → S, and h: S → S. The associative property states that (f ° g) ° h = f ° (g ° h). So if we apply an element x from set S to the composition (f ° g) ° h, it would be equivalent to applying x to the composition f ° (g ° h).

Therefore, the theorem that proves composition is associative in the set of one-to-one mappings from S to S is called the Associative Property of Composition.