Boolean Logic Question

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For anyone who knows anything about this, I have a question for you. This is an example of adding 2 byte long numbers. If the 2 digits being added are 1 and 1 with a carry out of 1, then the next carry in would be one. But if the next two digits were 0 and 0, the carry out would be zero, but for some reason the next carry in is still 1! I attached an image for anyone to see what I'm tlaking about. Can ya help me?!
Math was never my strong suit...

And you have just reminded me why.
Honestly, it has been so long since I did anything like that I can't remember how that works. I have taken symbolic logic and boolean equations but not in regards to adding bytes. I will have to look this stuff up and refresh the old grey matter...

Anyway, in looking at your table.... I see pattern sequences in the vertical columns that are consistent. And as I write this I have found a page that may be of assistance to you... How Boolean Logic Works I'll be giving this a good look over once I'm back home! Hope it helps:)
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