Construct the multiplication table modulo 8 for the set 1, 3, 5, 7 and evaluate (5 ullet 3) ullet 7. Then find the truth set of m ullet (m+2) = 3.

The operation $a \otimes b$ means $(a \times b) \pmod{8}$.

$$\begin{array}{|c|c|c|c|c|} \hline \otimes & 1 & 3 & 5 & 7 \\ \hline 1 & (1 \times 1) \pmod{8} = 1 & (1 \times 3) \pmod{8} = 3 & (1 \times 5) \pmod{8} = 5 & (1 \times 7) \pmod{8} = 7 \\ \hline 3 & (3 \times 1) \pmod{8} = 3 & (3 \times 3) \pmod{8} = 1 & (3 \times 5) \pmod{8} = 7 & (3 \times 7) \pmod{8} = 5 \\ \hline 5 & (5 \times 1) \pmod{8} = 5 & (5 \times 3) \pmod{8} = 7 & (5 \times 5) \pmod{8} = 1 & (5 \times 7) \pmod{8} = 3 \\ \hline 7 & (7 \times 1) \pmod{8} = 7 & (7 \times 3) \pmod{8} = 5 & (7 \times 5) \pmod{8} = 3 & (7 \times 7) \pmod{8} = 1 \\ \hline \end{array}$$

The completed table is:

$$\begin{array}{|c|c|c|c|c|} \hline \otimes & 1 & 3 & 5 & 7 \\ \hline 1 & 1 & 3 & 5 & 7 \\ \hline 3 & 3 & 1 & 7 & 5 \\ \hline 5 & 5 & 7 & 1 & 3 \\ \hline 7 & 7 & 5 & 3 & 1 \\ \hline \end{array}$$

First, find $5 \otimes 3$ from the table.
From the table, $5 \otimes 3 = 7$.
Next, substitute this value into the expression:

$$(5 \otimes 3) \otimes 7 = 7 \otimes 7$$

Now, find $7 \otimes 7$ from the table.
From the table, $7 \otimes 7 = 1$.
Therefore, $(5 \otimes 3) \otimes 7 = 1$.

The evaluation is $\boxed{1}$.

We need to test each value of $m$ from the set $\{1, 3, 5, 7\}$. Remember that $m+2$ is also taken modulo 8.

• For $m=1$:

$$1 \otimes (1+2) = 1 \otimes 3$$

From the table, $1 \otimes 3 = 3$. Since $3 = 3$, $m=1$ is in the truth set.

• For $m=3$:

$$3 \otimes (3+2) = 3 \otimes 5$$

From the table, $3 \otimes 5 = 7$. Since $7 \neq 3$, $m=3$ is not in the truth set.

• For $m=5$:

$$5 \otimes (5+2) = 5 \otimes 7$$

From the table, $5 \otimes 7 = 3$. Since $3 = 3$, $m=5$ is in the truth set.

• For $m=7$:

$$7 \otimes (7+2) = 7 \otimes 9$$

Since we are working modulo 8, $9 \equiv 1 \pmod{8}$. So,

$$7 \otimes 9 \equiv 7 \otimes 1$$

From the table, $7 \otimes 1 = 7$. Since $7 \neq 3$, $m=7$ is not in the truth set.

The truth set consists of all values of $m$ for which the equation holds.
The truth set is $\boxed{\{1, 5\}}$.