Derived Rules: Arrow Exchange and DeMorgan's

~(S&R)
1. S→~R by DeMorgan's
2. ~S∨~R by DeMorgan's
3. ~S→~R by Arrow Exchange
4. S→R by Arrow Exchange
5. Neither rule applies
~(G→H)
1. ~G∨~H by DeMorgan's
2. ~G&~H by Arrow Exchange
3. G&H by Arrow Exchange
4. G&~H by Arrow Exchange
5. Neither rule applies
~(K∨H)
1. K&~H by DeMorgan's
2. ~K&~H by DeMorgan's
3. ~K&H by Arrow Exchange
4. K&H by Arrow Exchange
5. Neither rule applies
~(~A→B)
1. ~~A&~B by DeMorgan's
2. ~~A&B by DeMorgan's
3. ~A&~B by Arrow Exchange
4. ~A&B by Arrow Exchange
5. Neither rule applies
~(H&J)
1. ~H∨~J by DeMorgan's
2. H∨~J by DeMorgan's
3. H&~J by Arrow Exchange
4. ~H&J by Arrow Exchange
5. Neither rule applies
~(D→~F)
1. D∨~~F by DeMorgan's
2. D→~~F by DeMorgan's
3. D&~~F by Arrow Exchange
4. D&~F by Arrow Exchange
5. Neither rule applies
~(W∨E)
1. ~W∨~E by DeMorgan's
2. ~W→~E by DeMorgan's
3. ~W&~E by DeMorgan's
4. ~W&E by Arrow Exchange
5. Neither rule applies
~(~~A→~B)
1. ~~A∨~B by DeMorgan's
2. ~A&~B by DeMorgan's
3. ~~A&~~B by Arrow Exchange
4. ~~A&~~~B by Arrow Exchange
5. Neither rule applies
~(~R&~S)
1. ~R∨S by DeMorgan's
2. ~~R∨~~S by DeMorgan's
3. ~~R→~~S by Arrow Exchange
4. ~R→S by Arrow Exchange
5. Neither rule applies
~(~D∨F)
1. ~~D&~F by DeMorgan's
2. ~D&~~F by DeMorgan's
3. ~~D&~~F by DeMorgan's
4. ~D→~F by Arrow Exchange
5. Neither rule applies
~(~K→M)
1. ~~K&~M by DeMorgan's
2. ~~K&M by Arrow Exchange
3. ~K&~M by Arrow Exchange
4. K→~M by Arrow Exchange
5. Neither rule applies
~~(H→J)
1. H→J by DeMorgan's
2. ~H&J by DeMorgan's
3. H→J by Arrow Exchange
4. H&~J by Arrow Exchange
5. Neither rule applies
~((E∨R)&T)
1. ~(E∨R)→T by DeMorgan's
2. (E∨R)→~T by DeMorgan's
3. ~(E∨R)∨~T by DeMorgan's
4. ~(E∨R)→~T by Arrow Exchange
5. Neither rule applies
~((~N&M)→K)
1. ~(~N&M)&~K by DeMorgan's
2. ~~(N&M)&K by DeMorgan's
3. ~(~N&M)&K by Arrow Exchange
4. (~N&M)&~K by Arrow Exchange
5. Neither rule applies
~((A∨B)∨C)
1. ~(A∨B)&C by DeMorgan's
2. ~(A∨B)&~C by DeMorgan's
3. (A∨B)&~C by Arrow Exchange
4. ~(~(A∨B)→C) by Arrow Exchange
5. Neither rule applies
~((G∨H)→J)
1. ~(G∨H)&J by DeMorgan's
2. ~(G∨H)&~J by DeMorgan's
3. (G∨H)&~J by Arrow Exchange
4. ~(G∨H)→~J by Arrow Exchange
5. Neither rule applies
~(~N&M)→K
1. ~(~N&M)&~K by DeMorgan's
2. ~(~N&M)∨~K by DeMorgan's
3. (~N&M)&~K by Arrow Exchange
4. ~(~N&M)→K by Arrow Exchange
5. Neither rule applies
~~(~(S∨~R)→~T)
1. ~(~S∨~R)&T by DeMorgan's
2. (S∨~~R)&T by Arrow Exchange
3. ~(S∨R)&~~T by Arrow Exchange
4. ~(S∨~R)→T by Arrow Exchange
5. Neither rule applies
~((~N∨~M)&~K)
1. ~(~N∨~M)∨~~K by DeMorgan's
2. ~(~N∨~M)&~K by DeMorgan's
3. (~N∨~M)∨~~K by Arrow Exchange
4. ~(~N∨~M)→~K by Arrow Exchange
5. Neither rule applies
~((R∨~S)→~~T)
1. ~(R∨~S)&~~~T by DeMorgan's
2. ~(R∨~S)∨~~T by DeMorgan's
3. (R∨~S)&~~~T by Arrow Exchange
4. (R∨~S)&~~T by Arrow Exchange
5. Neither rule applies
~((~A∨~B)∨~C)
1. ~(~A∨~B)&~~C by DeMorgan's
2. ~(~~A∨~~B)&~~C by DeMorgan's
3. (~A∨~B)→~~C by Arrow Exchange
4. (A&~~B)&C by Arrow Exchange
5. Neither rule applies
~D→~(F&~G)
1. ~D→~(F&~~G) by DeMorgan's
2. ~~D&~~(F&~G) by DeMorgan's
3. ~D→~~(F&~G) by Arrow Exchange
4. ~D&~~(F&~G) by Arrow Exchange
5. Neither rule applies
~(~R∨~(S&~T))
1. ~R&~~(S&~T) by DeMorgan's
2. ~~R&~~(S&~T) by DeMorgan's
3. ~~R&~~(S&~T) by Arrow Exchange
4. ~~R∨~~(S→~T) by Arrow Exchange
5. Neither rule applies
~(~G→~(H&~J))
1. ~~G∨~~(H&~J) by DeMorgan's
2. ~~G&~~(H&~~J) by DeMorgan's
3. ~G&~~(H&~J) by Arrow Exchange
4. G→~(H&~J) by Arrow Exchange
5. Neither rule applies
~(A→(~B&C))
1. A∨~~(~B&~C) by DeMorgan's
2. A&~(~B&~C) by DeMorgan's
3. A&~(~B&C) by Arrow Exchange
4. ~A&~(~B&C) by Arrow Exchange
5. Neither rule applies
~(M∨(~N&J))
1. ~M&(~~N&~J) by DeMorgan's
2. ~M&~(~N&J) by DeMorgan's
3. M&~(~N&J) by Arrow Exchange
4. ~M→~(~N&J) by Arrow Exchange
5. Neither rule applies
~~(~(R&S)∨~(T∨M))
1. ~~(R&S)&~~(T∨M) by DeMorgan's
2. ~~(R&S)∨~~(T∨M) by DeMorgan's
3. ~(R&S)&~~(T∨M) by Arrow Exchange
4. ~~(R&S)&~~(T∨M) by Arrow Exchange
5. Neither rule applies
~(~(R&S)∨~(T∨M))
1. ~~(R&S)&~~(T∨M) by DeMorgan's
2. ~~(R&S)∨~~(T∨M) by DeMorgan's
3. ~(R&S)&~~(T∨M) by Arrow Exchange
4. ~~(R&S)&~~(T∨M) by Arrow Exchange
5. Neither rule applies
~(A→((~B&C)∨~D))
1. A→(~(~B&C)&~D) by DeMorgan's
2. ~A∨~((~B&C)∨~D) by DeMorgan's
3. ~A&((~B&C)∨~D) by Arrow Exchange
4. A&~((~B&C)∨~D) by Arrow Exchange
5. Neither rule applies
~(((R&S)&T)&G)
1. (~(R&S)∨~T)&G by DeMorgan's
2. ~((R&S)&T)∨~G by DeMorgan's
3. ((R&S)&T)&~G by Arrow Exchange
4. ((R&S)&T)→G by Arrow Exchange
5. Neither rule applies

Derived Rules: Arrow Exchange and DeMorgan's

For each question decide whether DeMorgan's or Arrow Exchange may be applied and choose the answer that would result.