jmc

In a triangle, the lengths of any two sides must add up to a value larger than the third length's side. This is known as the Triangle Inequality. Keeping this in mind, we list out cases based on the length of the shortest side.

Case 1: shortest side has length $1$. Then the other two sides must have lengths $7$ and $7$. This leads to the set $\{1,7,7\}$.

Case 2: shortest side has length $2$. Then the other two sides must have lengths $6$ and $7$. This leads to the set $\{2,6,7\}$.

Case 3: shortest side has length $3$. Then the other two sides can have lengths $6$ and $6$ or $5$ and $7$. This leads to the sets $\{3,6,6\}$ and $\{3,5,7\}$.

Case 4: shortest side has length $4$. Then the other two sides can have lengths $5$ and $6$ or $4$ and $7$. This leads to the sets $\{4,5,6\}$ and $\{4,4,7\}$.

Case 5: shortest side has length $5$. Then the other two sides must have lengths $5$ and $5$. This leads to the set $\{5,5,5\}$.

Hence there are $\boxed{7}$ sets of non-congruent triangles with a perimeter of $15$ units.