a && !(b || a)Always true?
Always false?
True if and only if a is false?
True if and only if b is false?
True if and only if either a or b is true?
a && !(b || a)Apply De Morgan’s Law.
a && !b && !aReorder.
a && !a && !bGroup.
(a && !a) && !ba && !a is a contradiction.
Contradictions are always false.
false && !bfalse && x is false for all x.
falseTherefore the answer is, “2. Always false.”
Write a column for each variable.
Add a column for each sub expression in terms of a single operator and one of the expressions in a column to the left.
With n variables write 2n rows under the header.
In the first column write 2n-1 Fs and then 2n-1 Ts.
In the next column alternate 2n-2 Fs with 2n-2 Ts.
And so on until 2n-n (i.e. alternating single Fs and Ts.)
a && !(b || a)
| a | b | b || a | !(b || a) | a && !(b || a) |
|---|---|---|---|---|
| F | F | F | T | F |
| F | T | T | F | F |
| T | F | T | F | F |
| T | T | T | F | F |
a && !(b || a) |
Original |
a && !b && !a |
By De Morgan |
a && !a && !b |
By commutation |
(a && !a) && !b |
By grouping |
false && !b |
By contradiction |
false |
By definition of && |