Set notation for specific cartesian product set?

Question

If I have a set $\{(1, H), (2, C), (3, F), (4, Z), (5, S), (6, L) \}$ is there any way to express this with set builder notation? If not, is there any other way to express this mathematically?

Answer 1

Your set is not the cartesian product of two sets. To see why, suppose it were equal to $U \times V$ for some sets $U$ and $V$. Since $(1,H) \in U \times V$ and $(2,C) \in U \times V$, it follows that $1 \in U$, $2 \in U$, $H \in V$ and $C \in V$. But then $(1,C) \in U \times V$, for example, which is not an element of your set.

However, your is a subset of the cartesian product $\{1,2,3,4,5,6\} \times \{C,F,H,L,S,Z\}$

Another interpretation of your set is as the graph of a particular function $\{1,2,3,4,5,6 \} \to \{C,F,H,L,S,Z\}$, but to express it as such would require you to write out that function, which I don't think would make the expression any more concise.

Answer 2

{(1,H),(2,C),(3,F),(4,Z),(5,S),(6,L)}
= { x : x = (1,H) or x = (2,C) or x = (3,F) or x = (4,Z) or x = (5,S) or x = (6,L) }