What does !a&&(a||b) simplify to?

Question

I'm a little confused on !a&&(a||b). If I look at it directly and interpret it simply, it appears as if it is the same as

!a&&a or !a&&b

but this seems a little weird because since a can't be true and false, it would only be true if the latter was true. I also interpreted it like this

!a || a&&b

I don't really know how I came up with this one but it just looks more logical since there are no contradictions. Can anyone help me on this please?

Answer 1

Tautology Table

 | a | b | !a | a || b |  !a && (a || b)  | !a && b | [ !a && (a || b) ] <=> [!a && b]    |
 |---|---|----|--------|------------------|---------|-------------------------------------|
 | 0 | 0 |  1 |   0    |      0           |    0    |                   1                 |
 | 0 | 1 |  1 |   1    |      1           |    1    |                   1                 |
 | 1 | 0 |  0 |   1    |      0           |    0    |                   1                 |
 | 1 | 1 |  0 |   1    |      0           |    0    |                   1                 |
 

"Proof"

1. According to the Principle of Distributivity statement !a && (a || b) is equivalent to (!a && a) || (!a && b).

2. Accordiong to the Law of Non-Contradiction (!a && a) is equivalent to false

3. Putting it all together:

   !a && (a || b) <=> (!a && a) || (!a && b) <=> false || (!a && b) <=> !a && b

Answer 2

You can simplify it like this (!a && b) because in expression (!a && a || !a && b) the condition !a && a is always false

Answer 3

In Java like in most ¹ languages the unary ! has higher precedence than &&.

So !a&&(a||b) is (!a)&&(a||b)

You can represent the truth table of that expression using a Karnaugh map:

      | a = 0 | a = 1 |
------+-------+-------+
b = 0 |   0   |   0   |
------+-------+-------+
b = 1 |   1   |   0   |
------+-------+-------+

Now, it can easily be seen that the only true case is when (!a) && b.

So !a&&(a||b) is !a && b

---

¹_{See comments below.}

Answer 4

It simply means !a && b, a must be false and b must be true, for it to be true