adder in binary addition: XOR instead of OR gate
There are two outputs from a half-adder. These are sum and carry. Assuming A and B are the inputs, then their output tables look like this:
$$
\begin{align*}
{\begin{array}{c|c}
  { Sum }
&amp; {
    \begin{smallmatrix}
        B\\
        \begin{array}{cc}
            \overbrace{\begin{array}{cc}0 &amp; 1\end{array}}
        \end{array}
    \end{smallmatrix}
  }\\
\hline
  {
    \begin{smallmatrix}
        \begin{array}{r}
           A \left\{
               \begin{array}{c}
                 0\\
                 1
               \end{array}
             \right.\\
       \end{array}
    \end{smallmatrix}
  }
&amp; {
     \begin{smallmatrix}
         \begin{array}{c}
             \left.\begin{array}{cc}0&amp;1\\1&amp;0\end{array}\right.\\
         \end{array}
     \end{smallmatrix}}
\end{array}}
&amp;&amp;&amp;&amp;
{\begin{array}{c|c}
  {Carry }
&amp; {
    \begin{smallmatrix}
        B\\
        \begin{array}{cc}
            \overbrace{\begin{array}{cc}0 &amp; 1\end{array}}
        \end{array}
    \end{smallmatrix}
  }\\
\hline
  {
    \begin{smallmatrix}
        \begin{array}{r}
           A \left\{
               \begin{array}{c}
                 0\\
                 1
               \end{array}
             \right.\\
       \end{array}
    \end{smallmatrix}
  }
 &amp; {
     \begin{smallmatrix}
         \begin{array}{c}
             \left.\begin{array}{cc}0&amp;0\\0&amp;1\end{array}\right.\\
         \end{array}
     \end{smallmatrix}
   }
\end{array}}
\end{align*}
$$
You could just put the above tables together into a single table that shows both the sum and the carry-out. Something like this:
$$
\begin{align*}
{\begin{array}{c|c}
  { }
&amp; {
    \begin{matrix}
        B\\
        \begin{array}{cc}
            \overbrace{\begin{array}{cc}0 &amp; 1\end{array}}
        \end{array}
    \end{matrix}
  }\\
\hline
  {
    \begin{matrix}
        \begin{array}{r}
           A \left\{
               \begin{array}{c}
                 0\\
                 1
               \end{array}
             \right.\\
       \end{array}
    \end{matrix}
  }
&amp; {
     \begin{matrix}
         \begin{array}{c}
             \left.\begin{array}{cc}^00&amp;^01\\^01&amp;^10\end{array}\right.\\
         \end{array}
     \end{matrix}}
\end{array}}
\end{align*}
$$
Either way you draw it, the result is the same. The sum has the same truth table as an XOR does and the carry has the same truth table as an AND does.
It's that simple.
Don't get &quot;mired&quot; into words. Look at the behavior, not the words people use for things. You can know the name of a bird in every language on Earth and not know a single thing about the bird. The way you learn about the bird is to watch it and see how it behaves.
If someone says OR is (+), that's just a symbol that is supposed to imply a meaning to you -- but only if you've been trained about that meaning. In this case, it means inclusive as in, &quot;If A is true, or if B is true, or if both A and B are true, then the result is true.&quot; Please note that this includes a special case where both are true! So the inclusive-OR, aka (+), is a little different than addition.