You can use the [code]...[/code] tags for alignment because they preserve spaces.
"Valid" can be said about formulas, not truth tables.
A truth table for a formula with 3 variables should have 8 rows.
Hello, I've been working on truth tables recently and have come across the phrase 'validity', For the last day or so I've been trying to figure out how to calculate the validity of truth tables and I've come up with this below. (p.s I was unsure if this is the right forum section or not, so if not I'm sorry for that.)
if A > C & B > C, then (A or B) > C
| A | C | A > C |.... B.......|.......... B > C ........... |.. A > C & B > C ..| .....(A or B) > C
---------------------------------- -------------------------- ----------------------- ------
| T | T |.....T....| .. .T. .. |..............T...............| .............T............. .| ..............T..............
| T | F |.....F....| .. .T. .. |..............F................| ..............F..............| ..............F...........
| F | T |.....T....| .. .F. .. |..............T................| ..............T..............| ..............T...........
| F | F |.....T....| .. .F. .. |..............T................| ..............T..............| ..............T..........
I believe this to be a valid truth table, because when the premise is true, the final expression is true, and when the premise is false, so is the final expression. Can anyone confirm/deny and explain why.
many thanks to all that reply.
A -> C is F for A = T and C = F (4th row). The formula A > C & B > C > C requires parentheses. First, why is there "> C" in the end? Without parentheses, if & binds stronger than > and > associates to the right (which is the usual convention) A > C & B > C is parsed as A > ((C & B) > C). Check the value of (A > C) & (B > C) when A = C = F, B = T and when A = C = T and B = F.
The truth table is correct. For the argument to be valid, it is not necessary to for the conclusion to be false when the premise is false; it is only necessary for the conclusion to be true when the premise is true. Here, the formulas (A > C) & (B > C) and (A or B) > C are, in fact, equivalent, so their truth values always coincide.
I also recommend understanding why these formulas are equivalent using deduction rather than truth tables. Going left to right, we assume A > C, B > C and (A or B). If A, then we get C using A > C, and if B, we get C using B > C. You can similarly go right to left.