Question 1190026: its Contemporary Math
Construct a truth table for each of the following.
5) (š š ~š) ā (š ā§ ~š)
Found 2 solutions by math_tutor2020, MathLover1:
Answer by math_tutor2020(3817)   (Show Source):
You can put this solution on YOUR website!
T = true
F = false
We have three variables p, q and r
We need exactly  rows to go through all possible combos of truth values.
The first column for p will have four T's in a row followed by four F's in a row.
The second column for q will have two copies of TTFF
The third column for r will have four copies of TF
This will guarantee to capture every possible outcome
We have this table so far
| p | q | r | | T | T | T | | T | T | F | | T | F | T | | T | F | F | | F | T | T | | F | T | F | | F | F | T | | F | F | F |
Next is to add the column titled ~q which will just flip whatever is in column q
| p | q | r | ~q | | T | T | T | F | | T | T | F | F | | T | F | T | T | | T | F | F | T | | F | T | T | F | | F | T | F | F | | F | F | T | T | | F | F | F | T |
Do a similar operation for column ~r
| p | q | r | ~q | ~r | | T | T | T | F | F | | T | T | F | F | T | | T | F | T | T | F | | T | F | F | T | T | | F | T | T | F | F | | F | T | F | F | T | | F | F | T | T | F | | F | F | F | T | T |
Let's add the p ^ ~q column to the table we're building up
| p | q | r | ~q | ~r | p ^ ~q | | T | T | T | F | F | F | | T | T | F | F | T | F | | T | F | T | T | F | T | | T | F | F | T | T | T | | F | T | T | F | F | F | | F | T | F | F | T | F | | F | F | T | T | F | F | | F | F | F | T | T | F |
Recall that p ^ q is only true when p,q are true at the same time; otherwise, it's false.
Next, we can form the q v ~r column
| p | q | r | ~q | ~r | p ^ ~q | q v ~r | | T | T | T | F | F | F | T | | T | T | F | F | T | F | T | | T | F | T | T | F | T | T | | T | F | F | T | T | T | T | | F | T | T | F | F | F | T | | F | T | F | F | T | F | T | | F | F | T | T | F | F | F | | F | F | F | T | T | F | T |
The disjunction p v q is only false when p,q are both false; otherwise, it's true.
And finally, here is the table that you'll have as the final answer.
| p | q | r | ~q | ~r | p ^ ~q | q v ~r | (q v ~r) -> (p ^ ~q) | | T | T | T | F | F | F | T | F | | T | T | F | F | T | F | T | F | | T | F | T | T | F | T | T | T | | T | F | F | T | T | T | T | T | | F | T | T | F | F | F | T | F | | F | T | F | F | T | F | T | F | | F | F | T | T | F | F | F | T | | F | F | F | T | T | F | T | F |
It's definitely a lot to take in, especially because of the 8 rows to keep track of everything.
For the last column, the statement p -> q is only false when p = T and q = F; otherwise, the conditional statement is true.
Answer by MathLover1(20850)  (Show Source):
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