Simplify $$$0 \oplus 1$$$

Apply the formula $$$x \oplus y = \left(x \cdot \overline{y}\right) + \left(\overline{x} \cdot y\right)$$$ with $$$x = 0$$$ and $$$y = 1$$$:

$${\color{red}\left(0 \oplus 1\right)} = {\color{red}\left(\left(0 \cdot \overline{1}\right) + \left(\overline{0} \cdot 1\right)\right)}$$

Apply the negation law $$$\overline{1} = 0$$$:

$$\left(0 \cdot {\color{red}\left(\overline{1}\right)}\right) + \left(\overline{0} \cdot 1\right) = \left(0 \cdot {\color{red}\left(0\right)}\right) + \left(\overline{0} \cdot 1\right)$$

Apply the negation law $$$\overline{0} = 1$$$:

$$\left(0 \cdot 0\right) + \left({\color{red}\left(\overline{0}\right)} \cdot 1\right) = \left(0 \cdot 0\right) + \left({\color{red}\left(1\right)} \cdot 1\right)$$

Apply the dominant (null, annulment) law $$$x \cdot 0 = 0$$$ with $$$x = 0$$$:

$${\color{red}\left(0 \cdot 0\right)} + \left(1 \cdot 1\right) = {\color{red}\left(0\right)} + \left(1 \cdot 1\right)$$

Apply the commutative law:

$${\color{red}\left(0 + \left(1 \cdot 1\right)\right)} = {\color{red}\left(\left(1 \cdot 1\right) + 0\right)}$$

Apply the identity law $$$x + 0 = x$$$ with $$$x = 1 \cdot 1$$$:

$${\color{red}\left(\left(1 \cdot 1\right) + 0\right)} = {\color{red}\left(1 \cdot 1\right)}$$

Apply the identity law $$$x \cdot 1 = x$$$ with $$$x = 1$$$:

$${\color{red}\left(1 \cdot 1\right)} = {\color{red}\left(1\right)}$$