Q)In the text we defined permutations of a set and permutations of a sequence. Consider the following alternative definition of a sequence $\sigma : I \to A$. Let S be the set of elements that appear in $\sigma$; more precisely, $S = \{x: \text{ for some } i \in I, x = \sigma(i) \}$. Then a permutation of $\sigma$ is a permutation of the set S. Is this definition equivalent to the definition of permutation of sequence given in the text. Justify your answer? \
Vocabulary:
A sequence is defined as a function (First n elements of N) $\to$ alphabet
A function f: $A \to B$ is defined as a relation on A and B
Definition the permutation given by the text:
Let A be a finite set. A permutation of A is a sequence in which every element of A appears once and only once. For example if $A = \{a, b, c, d \}$ then $(b, a, c , d), (a, c, d, b)$, and $(a, b, c, d)$ are permutations of A (24 total). \
Sometimes we speak permutations of a sequence (rather than a set). In this case, the definition is as follows $\sigma: I \to A$ and $\tau: I \to A$ be finite sequences over A (note that the two sequences have the same length, $|I|$). The sequence $\tau$ is a permutation of $\sigma$ if there is a bijective function $f: I \to I$ so that for every $i \in I, \tau(i) = \sigma(f(i))$
Solution:
This definition is not equivalent because $set \neq sequence$, if you take a permutation of a set $\{1,2,3\}$ is equivalent to $\{2,1,3\}$, while a sequence order matters, not really sure to be honest.
The issue here is not that a set is unordered. That's fine. The problem is that a sequence may have repeated elements, and this would be lost when passing to a set. If the sequence were an injective function, it would be equivalent.