Question

Translate the following statements into symbolic form using capital letters to represent affirmative
Identify the main operator in the following propositions:

~[~(P & ~Q)] & ~(R –>~S)
~[~L v ~(J –> ~M)]
~[(H & ~L) & ~B] –> ~(~Y & ~K)
~[~(~F v J) v (~M v ~E)] & ~(~C & ~L)

Answer 1

~[~(P & ~Q)] & ~(R –>~S)

$\iff$~[[~(P & ~Q)] v (R –>~S)]

$\iff$ ~[~(P & ~Q) v (~R v~S)] (Using DeMorgan's laws in each step)

$\iff$ ~[~(P & ~Q) v ~(R & S)]

$\iff$ (P & ~Q) & (R & S)

$\iff$ P & ~Q & R & S

~[~L v ~(J –> ~M)]

$\iff$ ~[~L v ~(~J v ~M)]

$\iff$ ~[~L v (J & M)]

$\iff$ ~[(~L v J) & (~L v M)] (Using DeMorgan's laws in each step)

$\iff$ ~[(L->J) & (L->M)] (Using definition of $p \to q \iff \lnot p v q$ )

$\iff$ ~(L->J) v ~(L->M)

~[(H & ~L) & ~B] –> ~(~Y & ~K)

$\iff$ (~Y & ~K) –> [(H & ~L) & ~B] (Using $p \to q \iff \lnot q \to \lnot p$ )

$\iff$ (~Y & ~K) –> [~(~H v L) & ~B] (Using DeMorgan's laws in each step)

$\iff$ ~(Y v K) –> ~[(~H v L) v B]

$\iff$ [(~H v L) v B] -> (Y v K) (Using $p \to q \iff \lnot q \to \lnot p$ )

$\iff$ [(H -> L) v B] -> (Y v K) (Using definition of $p \to q \iff \lnot p v q$ )

~[~(~F v J) v (~M v ~E)] & ~(~C & ~L)

$\iff$ [ (~F v J) & ~(~M v ~E)] & (C v L) (Using DeMorgan's laws in each step)

$\iff$ [ (~F v J) & (M & E)] & (C v L)

$\iff$ [ (F -> J) & (M & E)] & (C v L) (Using definition of $p \to q \iff \lnot p v q$)