## Callisto 3 years ago Boolean algebra How can I get $$a+b+\bar{a}$$ from $$a+b+\bar{a}\bar{b}$$?

1. anonymous

Try DeMorgan

2. Callisto

$a+b+\bar{a}\bar{b}$$=\overline{\bar{a}\bar{b}} + \bar{a}\bar{b}$$=1$*cough* I just get the answer directly!?

3. UnkleRhaukus

what is bar ?

4. ParthKohli

Does $$+$$ mean $$\wedge$$ and $$\bar{a}$$ =$$\neg{a}$$

5. Callisto

$\overline{a}$ = complement of a

6. ParthKohli

Okay, so $$\neg a$$ indeed.

7. ParthKohli

$a + b + \bar{a} \iff a \wedge b \wedge \neg a$It's zero . . .

8. Callisto

9. ParthKohli

So does $$+$$ stand for the OR operator?

10. Callisto

Yes.

11. ParthKohli

then it is 1 indeed since either a or neg(a) is 1 :-)

12. ParthKohli

What does $$ab$$ mean? And?

13. Callisto

Yes :| Don't you write in this way?

14. ParthKohli

You can write this both ways :-)

15. UnkleRhaukus

$a\lor b\lor \neg a\iff a\lor b\lor\neg a\land\neg b$ ?

16. ParthKohli

Yes, they're pretty much equivalent using truth tables.

17. Callisto

I can't read (don't understand) all those symbols!! :(

18. ParthKohli

Eh, it's easy once you are used to it.

19. ParthKohli

Apparently we're not even caring about $$c$$ in that expression.

20. UnkleRhaukus

truth table ?

21. Callisto

If ∨ = OR ¬ = NOT ∧ = AND Then, the way to get a∨b∨¬a⟺a∨b∨¬a∧¬b is what I want know :|

22. ParthKohli

Well, manipulation. I thought you wanted a proof.

23. ParthKohli

It's just that if b is positive then you have the whole thing true, and if it's false then definitely ¬a∧¬b is true. If it's not true, then definitely a is true, so the whole expression will be true. So we have both expressions equivalent.

24. UnkleRhaukus

$\begin{array}{|c|c|c|c|}\hline a&\bar a&b&\bar b&\bar a\bar b&a+b+\bar a& a+b+\bar a \equiv \bar a\bar b \\\hline\ \end{array}$

25. ParthKohli

Dangy, that's just a truth table. I was waiting to see the manipulations @UnkleRhaukus does.

26. Callisto

I can get a∨b∨¬a = 1 and a∨b∨¬a∧¬b =1 (so they must be equal), but I don't know how to get a∨b∨¬a from a∨b∨¬a∧¬b

27. ParthKohli

Because they have the same boolean values for each case.

28. Callisto

...... I know they are equal, but how to get a∨b∨¬a from a∨b∨¬a∧¬b without showing that they are both equal to 1 and without using truth table (I'm sorry :( )?

29. UnkleRhaukus

is $$a$$= 1 , $$\bar a$$= 0 ?

30. ParthKohli

Yes.

31. ParthKohli

@UnkleRhaukus $$\neg$$ lol

32. Callisto

If a = 1, then $$\bar{a}$$=0.