Chapter 1 Review Quiz
(A): T, and (B): T.
(A): T, and (B): T. (A): T, and (B): F. (A): F, and (B): T. (A): F, and (B): F.
Simplification, 1
Addition, 1 Simplification, 1 Simplification, 2 Addition, 2
Proposition 1: false. Proposition 2: true.
Propositions 1 and 2 are defined as follows: Proposition 1: ¬(12−3=9) Proposition 2: (12−3=9)∧(5+2=7) Select the correct truth value for the two propositions. Question 10 options: Proposition 1: false. Proposition 2: false. Proposition 1: true. Proposition 2: false. Proposition 1: false. Proposition 2: true. Proposition 1: true. Proposition 2: true.
Hypothetical syllogism
Question 86 options: Modus ponens Hypothetical syllogism Disjunctive syllogism Conditional identity
¬q∨(p∧w)
Question 88 options: ¬q∨(p∧w) ¬q (p∧w) ¬q∧(p∧w)
Resolution (Wrong)
Resolution Hypothetical syllogism Modus ponens Modus tollens
q→p
Select the converse of p→q. Question 18 options: p→q q→p ¬p→¬q ¬q→¬p
If the train is late, then it rained this morning.
Select the converse of the following statement:If it rained this morning, then the train will be late. Question 25 options: If it didn't rain this morning, then the train will not be late. The train is not late despite the fact that it rained this morning.If the train is not late, then it didn't rain this morning.If the train is late, then it rained this morning.
If it didn't rain this morning, then the train will not be late.
Select the inverse of the following statement:If it rained this morning, then the train will be late. Question 24 options: If it didn't rain this morning, then the train will not be late. The train is not late despite the fact that it rained this morning. If the train is not late, then it didn't rain this morning. If the train is late, then it rained this morning.
Associative law (Wrong)
Select the law that shows that the two propositions are logically equivalent. (¬p∧(r∨¬q))∨(¬p∧w) and ¬p∧((r∨¬q)∨w) Question 42 options: DeMorgan's law Distributive law Associative law Commutative law
Associative law (Wrong)
Select the law that shows that the two propositions are logically equivalent. ¬((w∨p)∧(¬q∧q∧w)) and ¬(w∨p)∨¬(¬q∧q∧w) Question 41 options: DeMorgan's law Distributive law Associative law Complement law
Associative law (Wrong)
Select the law which shows that the two propositions are logically equivalent. (r∧p)∨(¬p∧q) and ((r∧p)∨¬p)∧((r∧p)∨q) Question 44 options: DeMorgan's law Distributive law Associative law Commutative law
DeMorgan's law
Select the law which shows that the two propositions are logically equivalent. (r∨q)∧(¬p∧¬q) and (r∨q)∧¬(p∨q) Question 46 options: DeMorgan's law Distributive law Associative law Commutative law
Absorption law
Select the law which shows that the two propositions are logically equivalent. (r∨q)∨((r∨q)∧p) and r∨q Question 45 options: Idempotent law Absorption law Identity law Domination law
Commutative law
Select the law which shows that the two propositions are logically equivalent. r∧(p∨q) and r∧(q∨p) Question 43 options: DeMorgan's law Distributive law Associative law Commutative law
∃x((¬S(x)∧¬Q(x))∨P(x))
Select the logical expression that is equivalent to: ¬∀x((S(x)∨Q(x))∧¬P(x)) Question 74 options: ∃x(¬(S(x)∧Q(x))∧P(x)) ∃x((¬S(x)∧¬Q(x))∨P(x)) ∃x((¬S(x)∧¬Q(x))∧P(x)) ∃x((¬S(x)∨¬Q(x))∧P(x))
∃x∀y(¬P(x)∨¬Q(x,y))
Select the logical expression that is equivalent to: ¬∀x∃y(P(x)∧Q(x,y)) Question 78 options: ∃x∀y(¬P(x)∨¬Q(x,y)) ∃y∀x(¬P(x)∨Q(x,y)) ∀y∃x(¬P(x)∨¬Q(x,y)) ∀x∃y(¬P(x)∨¬Q(x,y))
∀x(¬P(x)∨¬Q(x))
Select the logical expression that is equivalent to: ¬∃x(P(x)∧Q(x)) Question 68 options: ∃x(¬P(x)∨¬Q(x)) ∃x(¬P(x)∧¬Q(x)) ∀x(¬P(x)∨¬Q(x)) ∀x(¬P(x)∧¬Q(x))
∀x(¬P(x)∧(¬Q(x)∨¬S(x)))
Select the logical expression that is equivalent to: ¬∃x(P(x)∨(Q(x)∧S(x))) Question 72 options: ∀x(¬P(x)∨(¬Q(x)∨¬S(x))) ∀x(¬P(x)∨¬(Q(x)∧S(x))) ∀x(¬P(x)∧(¬Q(x)∨¬S(x))) ∀x(¬P(x)∨(¬Q(x)∧¬S(x)))
∀x(¬Q(x)∨(P(x)∧S(x))) (wrong)
Select the logical expression that is equivalent to: ¬∃x(Q(x)∧¬(P(x)∨S(x))) Question 73 options: ∀x(Q(x)∨P(x)∨S(x)) ∀x(¬Q(x)∨(P(x)∧S(x))) ∀x(¬Q(x)∨(P(x)∨S(x))) ∀x(¬Q(x)∨(¬P(x)∧¬S(x)))
∀x∃y¬(A(x)∨B(x,y)) (Wrong)
Select the logical expression that is equivalent to: ¬∃x∀y(A(x)∧B(x,y)) Question 82 options: ∀x∃y¬(A(x)∧B(x,y)) ∀x∃y(¬A(x)∧¬B(x,y)) ∀y∃x¬(A(x)∧B(x,y)) ∀x∃y¬(A(x)∨B(x,y))
∃x(P(x)∧¬Q(x))
Select the logical expression that is equivalent to: ¬∀x(¬P(x)∨Q(x)) Question 69 options: ∃x(P(x)∧¬Q(x)) ∃x(¬P(x)∨Q(x)) ∀x(P(x)∨¬Q(x)) ∀x(¬P(x)∧Q(x))
¬(p∨q)∧p
Select the proposition that is a contradiction. Question 31 options: ¬(p∨q)∧p (p∨q)∧p (¬p∧q)↔p (¬p∧¬q)→p
(p∧q)→p
Select the proposition that is a tautology. Question 30 options: (p∧q)→¬p (p∨q)→p (p∧q)↔p (p∧q)→p
(p∧q)∨(¬p∧¬q)
Select the proposition that is logically equivalent to p↔q. Question 39 options: (p∨q)∧(¬p∨¬q) (p∧q)∨(¬p∧¬q) (p→q)∨(q→p) (¬p→q)∧(¬q→p)
¬(q→p) (Wrong)
Select the proposition that is logically equivalent to p∧¬q. Question 40 options: ¬(q→p) ¬(p→q) ¬(¬p→q) ¬(q→¬p)
p∧¬q (Wrong)
Select the proposition that is logically equivalent to ¬p→q. Question 32 options: p∧¬q p∨q ¬p∨q ¬p∧q
(p∨q)→q
Select the proposition that is neither a tautology or a contradiction. Question 37 options: (p∧q)→q (p∨q)→q q→(p∨q) (p∧¬p)→q
(¬p↔¬q)∧(p↔q)
Select the proposition that is neither a tautology or a contradiction. Question 38 options: (p↔¬q)∨(p↔q) (¬p↔¬q)∧(p↔q) (p↔¬q)∧(p↔q) (p→q)∨(q→p)
If 3 is a prime number, then 6 is a prime number.
Select the statement that is false. Question 20 options: If 3 is a prime number, then 5 is a prime number. If 4 is a prime number, then 6 is a prime number. If 4 is a prime number, then 5 is a prime number. If 3 is a prime number, then 6 is a prime number.
Take out the trash.
Select the statement that is not a proposition. Question 1 options: 5 + 4 = 88 It will be sunny tomorrow. Take out the trash. Chocolate is the best flavor.
Is it your birthday today?
Select the statement that is not a proposition. Question 5 options: The stock market will crash next month. January has 30 days. 5 x 4 = 25 Is it your birthday today?
Disjunctive syllogism
Simplification Hypothetical syllogism Disjunctive syllogism Resolution
∀x(x2>x)
The domain for variable x is the set of all integers. Select the statement that is false. Question 49 options: ∀x(x2≠5) ∀x(x2≥x) ∀x(x2>x) ∃x(x=x)
∃x(x^2<1)
The domain for variable x is the set of all integers. Select the statement that is true. Question 50 options: ∃x(3x=1) ∃x(x^2<1) ∀x(x^2=1) ∃x(x^2<0)
∃x(3x=0) (Wrong)
The domain for variable x is the set of all positive real numbers. Select the statement that is true. Question 56 options: ∃x(3x=0) ∀x(x<2x) ∀x(x≠1) ∃x(x^2≤0)
∃x(x^2=−1)
The domain for variable x is the set of all real numbers. Select the statement that is false. Question 55 options: ∀x(x^2≥0) ∃x(x/2>x) ∃x(x^2=−1) ∃x(x^2=3)
∀x((¬N(x)∨¬R(x))→¬J(x))
The domain for variable x is the set of applicants for employment. Define the predicates: R(x): x brought a resume to the interview N(x): x arrived on time for the interview J(x): x got the job Select the logical expression that is equivalent to: "Every applicant who did not come on time for the interview or did not bring a resume did not get the job." Question 62 options: ∀x(¬N(x)→(¬R(x)∧¬J(x))) ∀x((¬N(x)∨¬R(x))→¬J(x)) ∀x(¬N(x)∧¬R(x)∧¬J(x)) ∀x((¬N(x)∨¬R(x))∧¬J(x))
∃x(J(x)→(¬N(x)∧¬R(x))) (Wrong)
The domain for variable x is the set of applicants for employment. Define the predicates: R(x): x brought a resume to the interview N(x): x arrived on time for the interview J(x): x got the job Select the logical expression that is equivalent to: "Someone got the job but did not come on time for the interview and did not bring a resume to the interview" Question 61 options: ∃x(J(x)∧(¬N(x)∨¬R(x))) ∃x(J(x)∧¬N(x)∧¬R(x)) ∃x(J(x)→(¬N(x)∨¬R(x))) ∃x(J(x)→(¬N(x)∧¬R(x)))
∃x(¬R(x)∧¬N(x)) (Wrong)
The domain for variable x is the set of applicants for employment. Define the predicates: R(x): x brought a resume to the interview N(x): x arrived on time for the interview Select the logical expression that is equivalent to: "At least one person did not bring a resume and at least one person did not come on time for the interview." Question 63 options: ∃x(¬R(x)∧¬N(x)) ∃x¬R(x)∧∃x¬N(x) ∃x(¬R(x)→¬N(x)) ∃x¬R(x)→∃x¬N(x)
The proposition is false.
The domain for variable x is the set of negative real numbers. Select the statement that correctly describes the proposition ∃x(x^2≤x). Question 53 options: The proposition is false. The proposition is true, and x = -1/2 is an example. The proposition is true, and x = 2 is an example. The proposition is true, and x = -2 an example.
The proposition is false, and x = 1/2 is a counterexample.
The domain for variable x is the set of positive real numbers. Select the statement that correctly describes the proposition ∀x(x^2≥x). Question 52 options: The proposition is true. The proposition is false, and x = 1/2 is a counterexample. The proposition is false, and x = 1 is a counterexample. The proposition is false, and x = -1 is a counterexample.
∀x(S(x)→A(x))
The domain for variable x is the set of students in a class. Define the predicates: S(x): x studied for the test A(x): x received an A on the test Select the logical expression that is equivalent to: "Everyone who studied for the test received an A on the test." Question 57 options: ∀x(A(x)→S(x)) ∀x(S(x)→A(x)) ∀x(S(x)∧A(x)) ∀x(S(x)↔A(x))
∃x(¬S(x)→A(x)) (Wrong)
The domain for variable x is the set of students in a class. Define the predicates: S(x): x studied for the test A(x): x received an A on the test Select the logical expression that is equivalent to: "Someone who did not study for the test received an A on the test." Question 58 options: ∃x(A(x)→¬S(x)) ∃x(¬S(x)→A(x)) ∃x(¬S(x)∧A(x)) ∃x(¬S(x)↔A(x))
∃x∀y(x+y≥0)
The domain for x and y is the set of real numbers. Select the statement that is false. Question 75 options: ∀x∃y(x+y≥0) ∃x∀y(x+y≥0) ∀x∃y(xy≥0) ∃x∀y(xy≥0)
∀x∃y(x+y=2y) (Wrong)
The domain for x and y is the set of real numbers. Select the statement that is false. Question 79 options: ∀x∃y(x+y=2x) ∃x∀y(xy=x) ∀x∃y(x+y=2y) ∃x∀y(xy=y)
P(0, 2, 2)
The predicate P and Q are defined as: Q(x,y):(x+y)/2=4 P(x,y,z):(x+y)/2=z The domain for all the variables is the set of real numbers. Select the proposition that is false. Question 54 options: Q(4, 4) P(0, 2, 2) Q(0, 8) P(-2, 2, 0)
T(4, 1, 25)
The predicate T is defined as:T(x,y,z):(x+y)2=zSelect the proposition that is true. Question 51 options: T(4, 1, 5) T(4, 1, 25) T(1, 1, 1) T(4, 0 2)
b→(s∧¬v)
The propositional variables b, v, and s represent the propositions: b: Alice rode her bike today. v: Alice overslept today. s: It is sunny today. Select the logical expression that represents the statement: "Alice rode her bike today only if it was sunny today and she did not oversleep." Question 21 options: b→s→¬v b→(s∧¬v) s∧(¬v→b) (s∧¬v)→b
(p∧¬h)∨f (Wrong)
The propositional variables f, h, and p represent the propositions: f: The student got an A on the final. h: The student turned in all the homework. p: The student is on academic probation Select the logical expression that represents the statement: "The student is on academic probation and did not turn in all the homework but still got an A on the final." Question 16 options: p∧h∧¬f (p∧¬h)∨f p∧¬h∧f (p∧h)∨¬f
¬p∧(f∨h)
The propositional variables f, h, and p represent the propositions: f: The student got an A on the final. h: The student turned in all the homework. p: The student is on academic probation Select the logical expression that represents the statement: "The student is not on academic probation and the student got an A on the final or turned in all the homework." Question 13 options: ¬p∧(f∨h) (¬p∧f)∨h ¬p∧f∧h ¬(p∧f)∨h
¬m→e (Wrong)
The propositional variables m and e represent the propositions: m: The patient is allergic to the medicine. e: The patient is eligible to participate in the study. Select the logical expression that represents the statement: "It is necessary for the patient to not be allergic to the medicine in order for the patient to be eligible for the study." Question 28 options: e→¬m ¬m→e ¬e→m e→m
s∧m
The propositional variables s and m represent the two propositions: s: It is sunny today. m: I will bring my umbrella. Select the logical expression that represents the statement: "Despite the fact that it is sunny today, I will bring my umbrella."
p = T, and q = F
Under what conditions do the expressions p∧q and ¬p have the same truth value? Question 9 options: p = q = F p = F, and q = T p = T, and q = F p = q = T
Existential generalization
Universal instantiation Universal generalization Existential instantiation Existential generalization
Existential instantiation
Universal instantiation Universal generalization Existential instantiation Existential generalization
Universal generalization
Universal instantiation Universal generalization Existential instantiation Existential generalization
Universal instantiation
Universal instantiation Universal generalization Existential instantiation Existential generalization
The employee did not receive a big bonus or does not have a big office.
Use De Morgan's law to select the statement that is equivalent to: "It is not true that the employee received a large bonus and has a big office." Question 33 options: The employee received a big bonus or has a big office. The employee did not receive a big bonus and does not have a big office. The employee did not receive a big bonus or does not have a big office. The employee received a big bonus and has a big office.
The patient does not have high blood pressure and does not have influenza.
Use De Morgan's law to select the statement that is equivalent to: "It is not true that the patient has high blood pressure or influenza." Question 34 options: The patient has high blood pressure or has influenza. The patient does not have high blood pressure and does not have influenza. The patient does not have high blood pressure or does not have influenza. The patient has high blood pressure and has influenza.
It's not true that the employee has a big office or received a large bonus.
Use De Morgan's law to select the statement that is equivalent to: "The employee did not receive a big bonus and does not have a big office. " Question 35 options: The employee did not receive a big bonus or the employee does not have a big office. It's not true that the employee has a big office and received a large bonus. It's not true that the employee has a big office or received a large bonus. If the employee received a large bonus then the employee also has a big office.
It is not true that the patient has high blood pressure and influenza.
Use De Morgan's law to select the statement that is equivalent to: "The patient does not have high blood pressure or the patient does not have influenza." Question 36 options: It is not true that the patient has high blood pressure and influenza. It is not true that the patient has high blood pressure or influenza. The patient does not have high blood pressure and the patient does not have influenza. The patient has high blood pressure but not influenza.
There is a student who did not get an A on the test.
Use De Morgan's law to select the statement that is logically equivalent to: "It is not true that every student got an A on the test." Question 66 options: Every student did not get an A on the test. There is a student who got an A on the test. Every student got an A on the test. There is a student who did not get an A on the test.
There is a student who didn't turn in the lab and didn't turn in the written homework.
Use De Morgan's law to select the statement that is logically equivalent to: "It is not true that every student turned in the lab or the written homework." Question 71 options: Some student turned in the written homework but not the lab. Some student turned in the lab but not the written homework. There is a student who didn't turn in the lab or didn't turn in the written homework. There is a student who didn't turn in the lab and didn't turn in the written homework.
Every student didn't turn in the lab or didn't turn in the written homework.
Use De Morgan's law to select the statement that is logically equivalent to: "It is not true that there is a student who turned in the lab and the written homework." Question 70 options: Every student turned in the lab or the written homework. Every student who didn't turn in the lab also didn't turn in the written homework. Every student didn't turn in the lab and didn't turn in the written homework. Every student didn't turn in the lab or didn't turn in the written homework.
Every student was not absent yesterday.
Use De Morgan's law to select the statement that is logically equivalent to: "It is not true that there was a student who was absent yesterday." Question 67 options: Every student was absent yesterday. Every student was not absent yesterday. Some student was absent yesterday. At least one student was not absent yesterday.
(p∨¬q)∧(p∨¬r)≡p∨(¬q∧¬r)
Which logical equivalence is an example of the Distributive law? Question 47 options: (p∨¬q)∧(p∨¬r)≡p∨(¬q∧¬r) (p∧¬q)∨(p∨¬r)≡((p∧¬q)∨p)∨¬r (p∨¬q)∧(p∨¬r)≡(¬¬p∨¬q)∧(p∨¬r) (p∨¬q)∧(p∨¬r)≡(p∨¬r)∧(p∨¬q)
Existential generalization
Which rule is used in the argument below? Alice is a student in the class. Alice got an A on the test and did not study. Therefore, there is a student in the class who got an A on the test and did not study. Question 94 options: Universal instantiation Universal generalization Existential instantiation Existential generalization
Universal instantiation
Which rule is used in the argument below? Every employee who met their performance goals received a raise. Jose is an employee. Therefore, if Jose met his performance goals then Jose received a raise. Question 95 options: Universal instantiation Universal generalization Existential instantiation Existential generalization
Existential generalization
Which rule is used in the argument below? Jose is an employee. Jose worked hard and received a bonus. Therefore, there is an employee who worked hard and received a bonus. Question 98 options: Universal instantiation Universal generalization Existential instantiation Existential generalization
3x≠12, then x≠4.
Which statement is the contrapositive of: "If x=4, then 3x=12." Question 19 options: If x=4, then 3x=12. If 3x=12, then x=4. If n≠12, then 3x≠12. 3x≠12, then x≠4.
p
p q p∨q p∧q
¬q
p = F, q = T, and r = T. Select the expression that evaluates to false. Question 3 options: ¬q q∨r q∧r p∨r
(¬p∧r)∨q
p = F, q = T, and r = T. Select the expression that evaluates to true. Question 15 options: ¬(q∨r) (¬p∧r)∨q (¬q∨r)∧p p∨¬q∨¬r
(¬p∧r)→q
p = F, q = T, and r = T. Select the expression that evaluates to true. Question 23 options: ¬(q∨p)↔r (¬p∧r)→q (q∨¬r)→p q↔(p∧r)
The proposition is true, regardless of the truth value for r.
p = T and q = F. Select the correct statement about the proposition: (p∧q)→r Question 29 options: The proposition is true, regardless of the truth value for r. The proposition is false, regardless of the truth value for r. If r = T, then the propositions is true, and if r = F, then the proposition is false. If r = F, then the propositions is true, and if r = T, then the proposition is false.
¬(q∨r∨¬p)
p = T, q = F, and r = F. Select the expression that evaluates to true. Question 12 options: ¬(q∨r∨¬p) (q∨r)∧p r∨q∨¬p (p∨r)∧(q∨¬p)
p∨r
p = T, q = F, and r = F. Select the expression that evaluates to true. Question 2 options: p∧q ¬p q∨r p∨r
¬q
p = T, q = F, and r = F. Select the expression that evaluates to true. Question 7 options: ¬q r∨q q∧p p∧r
r↔(p∧¬q) = T, and p→¬(q∨r) = F
p = T, q = F, and r = T. Select the correct truth values for the two propositions: r↔(p∧¬q) and p→¬(q∨r) Question 26 options: r↔(p∧¬q) = T, and p→¬(q∨r) = T r↔(p∧¬q) = T, and p→¬(q∨r) = F r↔(p∧¬q) = F, and p→¬(q∨r) = T r↔(p∧¬q) = F, and p→¬(q∨r) = F
¬(p∧¬q)
p = T, q = F, and r = T. Select the expression that evaluates to false. Question 11 options: p∨¬q p∨q∨r ¬(p∧¬q) ¬(p∧q∧r)
(p∧r)→q
p = T, q = F, and r = T. Select the expression that evaluates to false. Question 22 options: ¬(q∧r)→p (p∧r)→q (q∧r)→p (q∧r)→¬p
(q∨r)→¬p = F, and ¬r↔(p∧q) = T
p = T, q = T, and r = F. Select the correct truth values for the two propositions: (q∨r)→¬p and ¬r↔(p∧q) Question 27 options: (q∨r)→¬p = T, and ¬r↔(p∧q) = T (q∨r)→¬p = T, and ¬r↔(p∧q) = F (q∨r)→¬p = F, and ¬r↔(p∧q) = T (q∨r)→¬p = F, and ¬r↔(p∧q) = F
¬(p∨q∨r)
p = T, q = T, and r = F. Select the expression that evaluates to false. Question 14 options: ¬r∨(p∧q) q∧¬r ¬(p∧r)∨r ¬(p∨q∨r)
q∧r
p = T, q = T, and r = F. Select the expression that evaluates to false. Question 6 options: ¬r p∨r q∧r p∨q
p∨q: true and, p⊕q: false
p = T, q = T. Select the correct truth value for the propositions p∨q and p⊕q. Question 8 options: p∨q: true, and p⊕q: true p∨q: true and, p⊕q: false p∨q: false, and p⊕q: true p∨q: false, and p⊕q: false
¬(p∧q)
¬p ¬(p∧q) ¬q ¬p∧¬q
¬(p∨q)
¬p ¬q ¬p∨¬q ¬(p∨q)
¬¬(q∨p) (Wrong)
¬¬q∨p q∨¬¬p ¬¬(q∨p) ¬q∨¬p
∀x(Q(x)→P(x))
∀x(Q(x)→P(x)) ∀x(P(x)→Q(x)) ∀x(P(x)∧Q(x)) ∀x(P(x)∨Q(x))
∀x¬P(x,1)
∀x∀y¬P(x,y) ∀x∀yP(x,y) ∀y¬P(1,y) ∀x¬P(x,1)
∀x(A(x)→B(x))
∃x(A(x)→B(x)) ∃x((x≠Darth)∧A(x)∧B(x)) ∀x((x≠Luke)→(A(x)∨B(x))) ∀x(A(x)→B(x))
∃x(¬P(x)∧Q(x))
∃x(P(x)→Q(x)) ∃x(P(x)∧Q(x)) ∃x(¬P(x)∧Q(x)) ∃x(P(x)∧¬Q(x))
∃x¬(A(x)∨¬B(x)) (Wrong)
∃x(¬A(x)∧B(x)) ∃x¬(A(x)∨¬B(x)) ∀x((x≠Darth)∧(A(x)∨B(x))) ∀x((x≠Han)→(A(x)↔B(x)))
∃y∀xP(x,y)
∃x∀yP(x,y) ∀x∃yP(x,y) ∃y∀xP(x,y) ∀y∃xP(x,y)
∃y¬P(1,y)
∃x∃yP(x,y) ∃yP(2,y) ∀yP(1,y) ∃y¬P(1,y)
∀y∃x¬P(x,y) (Wrong)
∃y∀x¬P(x,y) ∃x∀yP(x,y) ∀y∃x¬P(x,y) ∀x∃y¬P(x,y)
