Answer to Question #108461 in Discrete Mathematics for Cristal Carrasco

Question #108461
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)
1
Expert's answer
2020-04-09T14:00:15-0400

~[~(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")



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