Simple Proof table problem

For some reason I can't wrap my head around this one...

I am supposed to use truth tables to explain expressions like

x ⊕ 1 = x

x ⊕ 1 = xnot

x ⊕ 0 = x

etc, etc.

Obviously I can do something like

but that seems redundant and doesn't really explain how I came to that conclusion. Is there anyway I can expand this truth table to make it clear how I came my conclusion? Any help would be appreciated, thanks!

Re: Simple Proof table problem

Hey HelpMeWanda.

What does the ⊕ mean? Usually in binary mathematics its an XOR operator - is this the case here?

Re: Simple Proof table problem

That's right, it's XOR, sorry for not being clearer.

Re: Simple Proof table problem

The first expression is wrong but the other two are ok. Basically you should just construct a truth table which will have four possibilities and calculate c = a AND not(b) d = b AND not(a) and e = c OR d.

Re: Simple Proof table problem

Ok, I see what you're saying and I've been trying to do that, but I have trouble making the conclusion work. The problem is that whenever I add a second variable to the truth table, it makes it so x(not) has a TRUE value in a row that isn't XOR. I'm having trouble explaining this I think, so I will draw out my truth table and show what's wrong.

x | y | x(not) | y(not) | x(not) AND y | x AND y(not) | x(not) AND y OR x AND y(not) |

0 | 0 | 1 | 1 | 0 | 0 | 0 |

0 | 1 | 1 | 0 | 0 | 1 | 1 |

1 | 0 | 0 | 1 | 1 | 0 | 1 |

1 | 1 | 0 | 0 | 0 | 0 | 0 |

So going by this table, it would seem x⊕y doesn't equal x(not) like it is supposed to. The only way it works is if I use one variable, the problem being I don't know how to flesh out a 1 variable truth table that explains how I came to this conclusion.

Re: Simple Proof table problem

You have to remember that y is fixed in your questions. In the order of the questions given, you have y = 1, 1, 0. Substitute these in and see if you get a contradiction or not (you should get a contradiction for the first statement).

Re: Simple Proof table problem

Wow thank you so much, I think I get it now.

Just to make sure though, given the way these questions were asked, is this acceptable:

Sorry I think that's wrong... for some reason like I said I just can't wrap my head around this... I really appreciate your help and patience though.

Re: Simple Proof table problem

You can remove the duplicates which will give you two rows.

Re: Simple Proof table problem

Thank you so much! You're a life saver.