Definition:Ordered Tuple/Term

Definition

Let $n \in \N_{>0}$.

Let $\sequence {a_k}_{k \mathop \in \N^*_n}$ be an ordered tuple.

The ordered pair $\tuple {k, a_k}$ is called the $k$th term of the ordered tuple for each $k \in \N^*_n$.

Also defined as

Some treatments of this subject treat the $k$th term of an ordered tuple as just the element $a_k$.

However, this is an oversimplification which obscures some of the crucial detail of the definition of what an ordered tuple actually is.

Also see

• Definition:Ordered Tuple as Ordered Set

• Results about ordered tuples can be found here.

Sources

• 1965: Seth Warner: Modern Algebra ... (previous) ... (next): Chapter $\text {III}$: The Natural Numbers: $\S 18$: Induced $N$-ary Operations
• 1971: Robert H. Kasriel: Undergraduate Topology ... (previous) ... (next): $\S 1.15$: Sequences
• 1996: H. Jerome Keisler and Joel Robbin: Mathematical Logic and Computability ... (previous) ... (next): Appendix $\text{A}.10$: Finite Sequences