Propositional Logic : Comprehensive Review

If Martina is the author, then the book is fiction. But the book is nonfiction. Therefore Martina is not the author. Which inference rule is illustrated?

Select the correct answer. The options are: Modus tollens, Modus ponens, Simplification, Addition.

If the business declares bankruptcy, then all assets must be confiscated. The business declared bankruptcy. It follows that all assets must be confiscated. Which inference rule is illustrated?

Select the correct answer. The options are: Modus ponens, Modus tollens, Conjunction, Simplification.

The dog has a shiny coat and loves to bark. Consequently, the dog loves to bark. Which inference rule is illustrated?

Select the correct answer. The options are: Simplification, Conjunction, Addition, Modus ponens.

If Paul is a good swimmer, then he is a good runner. If Paul is a good runner, then he is a good biker. Therefore if Paul is a good swimmer, then he is a good biker. Which inference rule is illustrated?

Select the correct answer. The options are: Hypothetical syllogism, Modus ponens, Modus tollens, Conjunction.

If the car was involved in the hit-and-run, then the paint would be chipped. But the paint is not chipped. What conclusion can be reached?

Select the correct answer. The options are: The car was not involved in the hit-and-run., The car was involved in the hit-and-run., The paint is chipped., No conclusion can be reached..

Either the weather will turn bad or we will leave on time. If the weather turns bad, then the flight will be canceled. What conclusion can be reached?

Select the correct answer. The options are: Either the flight will be canceled or we will leave on time., The flight will be canceled., We will leave on time., The weather will turn bad..

If the bill was sent today, then you will be paid tomorrow. You will be paid tomorrow. What conclusion can be reached?

Select the correct answer. The options are: No conclusion can be reached., The bill was sent today., The bill was not sent today., You will not be paid tomorrow..

The grass needs mowing and the trees need trimming. If the grass needs mowing, then we need to rake the leaves. What conclusion can be reached?

Select the correct answer. The options are: The trees need trimming and we need to rake the leaves., The grass does not need mowing., We do not need to rake the leaves., No conclusion can be reached..

In the proof sequence of \(A∧(B→C)→(B→(A∧C))\): 1. \(A\) 2. \(B→C\) 3. \(B\) 4. \(C\) 5. \(A∧C\). What justifies step 4?

Select the correct answer. The options are: Steps 2 and 3, modus ponens, Steps 1 and 3, conjunction, Step 2, simplification, Steps 1 and 2, hypothetical syllogism.

In the proof sequence of \(A∧(B→C)→(B→(A∧C))\): 1. \(A\) 2. \(B→C\) 3. \(B\) 4. \(C\) 5. \(A∧C\). What justifies step 5?

Select the correct answer. The options are: Steps 1 and 4, conjunction, Steps 1 and 3, conjunction, Steps 2 and 4, modus ponens, Step 4, addition.

In the proof sequence of \(B∧[(B∧C)→A']∧(B→C)→A'\): 1. \(B\) 2. \((B∧C)→A'\) 3. \(B→C\) 4. \(C\) 5. \(B∧C\) 6. \(A'\). What justifies step 5?

Select the correct answer. The options are: Steps 1 and 4, conjunction, Steps 1 and 3, modus ponens, Steps 2 and 4, modus ponens, Step 4, addition.

In the proof sequence of \(B∧[(B∧C)→A']∧(B→C)→A'\): 1. \(B\) 2. \((B∧C)→A'\) 3. \(B→C\) 4. \(C\) 5. \(B∧C\) 6. \(A'\). What justifies step 6?

Select the correct answer. The options are: Steps 2 and 5, modus ponens, Steps 1 and 5, conjunction, Steps 3 and 4, modus ponens, Step 5, simplification.

In the proof sequence of \([A→(B∨C)]∧B'∧C'→A'\): 1. \(A→(B∨C)\) 2. \(B'\) 3. \(C'\) 4. \(B'∧C'\) 5. \((B∨C)'\) 6. \(A'\). What justifies step 5?

Select the correct answer. The options are: Step 4, De Morgan's laws, Step 4, implication, Steps 2 and 3, conjunction, Step 4, double negation.

In the proof sequence of \([A→(B∨C)]∧B'∧C'→A'\): 1. \(A→(B∨C)\) 2. \(B'\) 3. \(C'\) 4. \(B'∧C'\) 5. \((B∨C)'\) 6. \(A'\). What justifies step 6?

Select the correct answer. The options are: Steps 1 and 5, modus tollens, Steps 1 and 5, modus ponens, Steps 1 and 5, hypothetical syllogism, Step 5, simplification.

In the proof sequence of \(A'∧B∧[B→(A∨C)]→C\): 1. \(A'\) 2. \(B\) 3. \(B→(A∨C)\) 4. \(A∨C\) 5. \((A')'∨C\) 6. \(A'→C\) 7. \(C\). What justifies step 5?

Select the correct answer. The options are: Step 4, double negation, Step 4, implication, Step 4, De Morgan, Step 4, commutative.

In the proof sequence of \(A'∧B∧[B→(A∨C)]→C\): 1. \(A'\) 2. \(B\) 3. \(B→(A∨C)\) 4. \(A∨C\) 5. \((A')'∨C\) 6. \(A'→C\) 7. \(C\). What justifies step 6?

Select the correct answer. The options are: Step 5, implication, Step 5, De Morgan, Step 5, double negation, Step 5, commutative.

To prove \((A∨B')'∧(B→C)→(A'∧C)\), what is the result of applying De Morgan's laws to \((A∨B')'\)?

Select the correct answer. The options are: \(A'∧B\), \(A∧B'\), \(A'∨B\), \(A∨B'\).

To prove \(A'∧(B→A)→B'\), which rule is used to derive \(B'\) from \(A'\) and \(B→A\)?

Select the correct answer. The options are: Modus tollens, Modus ponens, Hypothetical syllogism, Addition.

To prove \((A→B)∧[A→(B→C)]→(A→C)\), using the deduction method, which rule derives \(B\) after adding \(A\) as a hypothesis?

Select the correct answer. The options are: Modus ponens from \(A→B\) and \(A\), Modus tollens from \(A→B\) and \(A\), Conjunction of \(A→B\) and \(A\), Simplification of \(A→B\).

To prove \([(C→D)→C]→[(C→D)→D]\), using the deduction method, which rule derives \(D\) after deriving \(C\)?

Select the correct answer. The options are: Modus ponens from \(C→D\) and \(C\), Modus tollens from \(C→D\) and \(C\), Hypothetical syllogism, Addition.

To prove \(A'∧(A∨B)→B\), which rule is used to derive \(B\) from \(A'\) and \(A∨B\)?

Select the correct answer. The options are: Disjunctive syllogism, Modus ponens, Hypothetical syllogism, Conjunction.

To prove \([A→(B→C)]∧(A∨D')∧B→(D→C)\), using the deduction method, which rule derives \(A\) from \(A∨D'\) and \(D\)?

Select the correct answer. The options are: Disjunctive syllogism, Modus ponens, Modus tollens, Addition.

To prove \((A'→B')∧B∧(A→C)→C\), which rule derives \(A\) from \(A'→B'\) and \(B\)?

Select the correct answer. The options are: Modus tollens, Modus ponens, Hypothetical syllogism, Addition.

To prove \((A→B)∧[B→(C→D)]∧[A→(B→C)]→(A→D)\), what is added as an additional hypothesis by the deduction method?

Select the correct answer. The options are: \(A\), \(B\), \(D\), \(C\).

To prove \([A→(B→C)]→[B→(A→C)]\), using the deduction method, which two hypotheses are added?

Select the correct answer. The options are: \(A\) and \(B\), \(B\) and \(C\), \(A\) and \(C\), \(A→B\) and \(B→C\).

To prove \((A∧B)→(A→B')'\), after adding \(A∧B\) as a hypothesis, what is the result of simplifying?

Select the correct answer. The options are: \(A\) and \(B\), \(A\) only, \(B\) only, \(A'∧B'\).

To prove \((A→C)∧(C→B')∧B→A'\), which rule derives \(C'\) from \(C→B'\) and \(B\)?

Select the correct answer. The options are: Modus tollens, Modus ponens, Hypothetical syllogism, Addition.

To prove \([A→(B∨C)]∧C'→(A→B)\), using the deduction method, which rule derives \(B\) from \(B∨C\) and \(C'\)?

Select the correct answer. The options are: Disjunctive syllogism, Modus ponens, Modus tollens, Conjunction.

What is the name of the rule \((P∨Q)∧P'→Q\)?

Select the correct answer. The options are: Disjunctive syllogism, Hypothetical syllogism, Contraposition, Exportation.

What is the name of the rule \((P→Q)→(Q'→P')\)?

Select the correct answer. The options are: Contraposition, Disjunctive syllogism, Self-reference, Exportation.

What is the name of the rule \((Q'→P')→(P→Q)\)?

Select the correct answer. The options are: Contraposition, Disjunctive syllogism, Self-reference, Inconsistency.

What is the name of the rule \(P→P∧P\)?

Select the correct answer. The options are: Self-reference, Exportation, Inconsistency, Distributive.

What is the name of the rule \(P∨P→P\)?

Select the correct answer. The options are: Self-reference, Exportation, Inconsistency, Distributive.

What is the name of the rule \([(P∧Q)→R]→[P→(Q→R)]\)?

Select the correct answer. The options are: Exportation, Self-reference, Inconsistency, Contraposition.

What is the name of the rule \(P∧P'→Q\)?

Select the correct answer. The options are: Inconsistency, Exportation, Self-reference, Distributive.

What is the name of the rule \(P∧(Q∨R)→(P∧Q)∨(P∧R)\)?

Select the correct answer. The options are: Distributive, Exportation, Inconsistency, Contraposition.

What is the name of the rule \(P∨(Q∧R)→(P∨Q)∧(P∨R)\)?

Select the correct answer. The options are: Distributive, Exportation, Inconsistency, Self-reference.

To prove \(A'→(A→B)\), which rule transforms \(A'∨B\) into \(A→B\)?

Select the correct answer. The options are: Implication, De Morgan, Double negation, Commutative.

To prove \((P→Q)∧(P'→Q)→Q\), which equivalence converts both implications to disjunctions?

Select the correct answer. The options are: Implication, De Morgan, Contraposition, Exportation.

To prove \((A'→B')∧(A→C)→(B→C)\), which rule derives \(B→A\) from \(A'→B'\)?

Select the correct answer. The options are: Contraposition, Implication, De Morgan, Exportation.

To prove \((A'→B)∧(B→C)∧(C→D)→(A'→D)\), which rule chains the implications together?

Select the correct answer. The options are: Hypothetical syllogism, Modus ponens, Contraposition, Inconsistency.

To prove \((A∨B)∧(A→C)∧(B→C)→C\), which rule is used to derive \(C\) from the disjunction and the two implications?

Select the correct answer. The options are: Disjunctive syllogism after converting the implications, Modus ponens only, Addition only, Exportation only.

To prove \((Y→Z')∧(X'→Y)∧[Y→(X→W)]∧(Y→Z)→(Y→W)\), what is added by the deduction method?

Select the correct answer. The options are: \(Y\), \(X\), \(Z\), \(W\).

To prove \((A∧B)∧(B→A')→(C∧B')\), which rule yields \(A'\) from \(B→A'\) and \(B\)?

Select the correct answer. The options are: Modus ponens, Modus tollens, Hypothetical syllogism, Addition.

To prove \((A∧B)'∧(C'∧A)'∧(C∧B')'→A'\), what is the first step typically applied to each negated conjunction?

Select the correct answer. The options are: De Morgan's laws, Modus ponens, Hypothetical syllogism, Addition.

To prove \((P∨(Q∧R))∧(R'∨S)∧(S→T')→(T→P)\), what is added by the deduction method?

Select the correct answer. The options are: \(T\), \(P\), \(Q\), \(S\).

If the program is efficient, it executes quickly. Either the program is efficient, or it has a bug. However, the program does not execute quickly. Therefore, it has a bug. Using \(E\), \(Q\), \(B\), what is the symbolic form?

Select the correct answer. The options are: \((E→Q)∧(E∨B)∧Q'→B\), \((E→Q)∧(E∧B)∧Q'→B\), \((E∨Q)∧(E→B)∧Q'→B\), \((E→Q)∧(E∨B)∧B'→Q\).

If Jane is more popular, then she will be elected. If Jane is more popular, then Craig will resign. Therefore, if Jane is more popular, she will be elected and Craig will resign. Using \(J\), \(E\), \(C\), what is the symbolic form?

Select the correct answer. The options are: \((J→E)∧(J→C)→(J→(E∧C))\), \((J→E)∧(E→C)→(J→(E∧C))\), \((J∧E)∧(J∧C)→(J→(E∧C))\), \((J→E)∧(J→C)→(E∧C)\).

If chicken is on the menu, then don't order fish, but you should have either fish or salad. So, if chicken is on the menu, have salad. Using \(C\), \(F\), \(S\), what is the symbolic form?

Select the correct answer. The options are: \((C→F')∧(C→(F∨S))→(C→S)\), \((C→F)∧(C→(F∨S))→(C→S)\), \((C∧F')∧(F∨S)→(C→S)\), \((C→F')∧(F∨S)→(C∧S)\).

The crop is good, but there is not enough water. If there is a lot of rain or not a lot of sun, then there is enough water. Therefore, the crop is good and there is a lot of sun. Using \(C\), \(W\), \(R\), \(S\), what is the symbolic form?

Select the correct answer. The options are: \(C∧W'∧((R∨S')→W)→(C∧S)\), \(C∧W∧((R∨S)→W)→(C∧S)\), \(C∨W'∧((R∨S')→W)→(C∨S)\), \(C∧W'∧((R∧S')→W)→(C∧S)\).

If the ad is successful, then the sales volume will go up. Either the ad is successful or the store will close. The sales volume will not go up. Therefore, the store will close. Using \(A\), \(S\), \(C\), what is the symbolic form?

Select the correct answer. The options are: \((A→S)∧(A∨C)∧S'→C\), \((A→S)∧(A∧C)∧S'→C\), \((A∨S)∧(A→C)∧S'→C\), \((A→S)∧(A∨C)∧C'→S\).

If DeWayne is not tall then Jayden is not DeWayne's brother. If DeWayne is tall then Trevor is DeWayne's brother. Therefore, if Jayden is DeWayne's brother, then Trevor is DeWayne's brother. Using \(D\), \(J\), \(T\), what is the symbolic form?

Select the correct answer. The options are: \((D'→J')∧(D→T)→(J→T)\), \((D'→J')∧(D→T)→(T→J)\), \((D→J')∧(D→T)→(J→T)\), \((D'→J)∧(D→T)→(J→T)\).

Russia was a superior power, and either France was not strong or Napoleon made an error. Napoleon did not make an error, but if the army did not fail, then France was strong. Hence, the army failed and Russia was a superior power. Using \(R\), \(F\), \(N\), \(A\), what is the symbolic form?

Select the correct answer. The options are: \(R∧(F'∨N)∧N'∧(A'→F)→(A'∧R)\), \(R∧(F∨N)∧N'∧(A'→F)→(A'∧R)\), \(R∧(F'∨N)∧N∧(A'→F)→(A'∧R)\), \(R∧(F'∨N)∧N'∧(A→F)→(A'∧R)\).

It is not the case that if electric rates go up, then usage will go down, nor is it true that either new power plants will be built or bills will not be late. Therefore, usage will not go down and bills will be late. Using \(R\), \(U\), \(P\), \(B\), what is the symbolic form?

Select the correct answer. The options are: \((R→U)'∧(P∨B')'→(U'∧B)\), \((R→U)∧(P∨B')→(U'∧B)\), \((R→U)'∧(P∨B)→(U'∧B)\), \((R→U)'∧(P∨B')'→(U∧B')\).

If Jose took the jewelry or Mrs. Krasov lied, then a crime was committed. Mr. Krasov was not in town. If a crime was committed, then Mr. Krasov was in town. Therefore, Jose did not take the jewelry. Using \(J\), \(L\), \(C\), \(T\), what is the symbolic form?

Select the correct answer. The options are: \(((J∨L)→C)∧T'∧(C→T)→J'\), \(((J∨L)→C)∧T∧(C→T)→J'\), \(((J∧L)→C)∧T'∧(C→T)→J'\), \(((J∨L)→C)∧T'∧(C→T)→J\).

If the birds are flying south and the leaves are turning, then it must be fall. Fall brings cold weather. The leaves are turning but the weather is not cold. Therefore, the birds are not flying south. Using \(B\), \(L\), \(F\), \(C\), what is the symbolic form?

Select the correct answer. The options are: \(((B∧L)→F)∧(F→C)∧(L∧C')→B'\), \(((B∧L)→F)∧(F→C)∧(L∧C)→B'\), \(((B∨L)→F)∧(F→C)∧(L∧C')→B'\), \(((B∧L)→F)∧(F→C)∧(L∧C')→B\).

If a Democrat is elected then taxes will go up. Either a Democrat will be elected or the bill will pass. Therefore, if taxes do not go up, then the bill will pass. Using \(D\), \(T\), \(B\), what is the symbolic form?

Select the correct answer. The options are: \((D→T)∧(D∨B)→(T'→B)\), \((D→T)∧(D∧B)→(T'→B)\), \((D→T)∧(D∨B)→(T→B)\), \((D→T)∧(B→D)→(T'→B)\).

Either Emily was not home or if Pat did not leave the tomatoes, then Sophie was ill. Also, if Emily was not home, then Olivia left the peppers. But it is not true that either Sophie was ill or Olivia left the peppers. Therefore, Pat left the tomatoes and Olivia did not leave the peppers. Using \(E\), \(P\), \(S\), \(O\), what is the symbolic form?

Select the correct answer. The options are: \((E'∨(P'→S))∧(E'→O)∧(S∨O)'→(P∧O')\), \((E∨(P'→S))∧(E'→O)∧(S∨O)'→(P∧O')\), \((E'∨(P'→S))∧(E'→O)∧(S∨O)'→(P'∧O)\), \((E'∨(P→S))∧(E'→O)∧(S∨O)'→(P∧O')\).

Which equivalence justifies the deduction method that says: to prove \(P_1∧P_2∧...∧P_n→(R→S)\), deduce \(S\) from \(P_1,P_2,...,P_n\) and \(R\)?

Select the correct answer. The options are: \(A→(B→C)↔(A∧B)→C\), \(A→(B→C)↔(A∨B)→C\), \(A→(B→C)↔A∧B∧C\), \(A→(B→C)↔A→B→C\).

The defense attorney argues: If my client is guilty, then the knife was in the drawer. Either the knife was not in the drawer or Jason Pritchard saw the knife. If the knife was not there on October 10, it follows that Jason Pritchard didn't see the knife. Furthermore, if the knife was there on October 10, then the knife was in the drawer and also the hammer was in the barn. But we all know that the hammer was not in the barn. Therefore, my client is innocent. Using \(G\) (guilty), \(K\) (knife in drawer), \(J\) (Jason saw knife), \(T\) (knife there on Oct 10), \(H\) (hammer in barn), what is the symbolic form?

Select the correct answer. The options are: \((G→K)∧(K'∨J)∧(T'→J')∧(T→(K∧H))∧H'→G'\), \((G→K)∧(K∨J)∧(T'→J')∧(T→(K∧H))∧H'→G'\), \((G→K)∧(K'∨J)∧(T→J')∧(T→(K∧H))∧H'→G'\), \((G→K)∧(K'∨J)∧(T'→J')∧(T→(K∨H))∧H'→G'\).

In the defense attorney argument, what rule derives \(T'\) from \(T→(K∧H)\) and \(H'\)?

Select the correct answer. The options are: Modus tollens after obtaining \((K∧H)'\) from \(H'\), Modus ponens from \(T→(K∧H)\) and \(T\), Hypothetical syllogism, Addition.