i had a puzzle and used a logical argument to show a point but some says that my argument is wrong but i can't understand the reason they provide !
the puzzles says , Given four cards laid out on a table as: $D , 3 , F , 7$ , where each card has a letter on one side and a number on the other. then Which cards should you flip over to determine if every card with a $D$ on one side has a $7$ on the other side?
i solved so , my question is not to solve the puzzle i claimed that there is no need to flip $7$ over . and my argument as follows
let $P$ = the card has $D$ on one side $Q$ = the card has $7$ on the other side
let , $A$ $=$ $P$ $\rightarrow$ $Q$
$B$ $=$ $¬Q$ $\rightarrow$ $¬P$
from the truth table we know that any wff of those A and B tautolofically implies the other so they are equvlant and we can use any one of then instead of the other
so , we want to show that , if the card has $D$ on one side then it has $7$ on the other side
so we can use the equivlant wff which says , if the card doesn't have $7$ then it hasn't $D$ on the other side
and we know that the fourth card has $7$ , so $¬Q$ is wrong so $B$ so true so $A$ is true
so we don't need to flip card 7
is this argument right ?
they say that i have to show that $ A$ is true before using the equivlant between $A$ and $B $
what is right and why ?