Predicate Symbols, Arity, And Open Statements Quiz

1) “x does not satisfy P” is:

P(¬x)

¬P(x)

P(x) → False

¬x

The statement "x is not even" is the negation of "x is even". Since P(x) means "x is even", we write the negation as ¬P(x). The other options are incorrect: P(¬x) would mean "the negation of x is even", which doesn't make sense.

Explanation

The statement "x is not even" is the negation of "x is even". Since P(x) means "x is even", we write the negation as ¬P(x). The other options are incorrect: P(¬x) would mean "the negation of x is even", which doesn't make sense.

2) The predicate R(x,y) takes how many arguments?

0

1

2

Depends on domain

The predicate R(x,y) has two arguments: x and y. The number of arguments a predicate takes is called its arity, and it is fixed by the definition of the predicate. In this case, R is defined to take two arguments, so the arity is 2, regardless of the domain.

Explanation

The predicate R(x,y) has two arguments: x and y. The number of arguments a predicate takes is called its arity, and it is fixed by the definition of the predicate. In this case, R is defined to take two arguments, so the arity is 2, regardless of the domain.

3) Which is NOT an open predicate?

X < 4

X + y = 7

5 is odd

Y > 2

An open predicate is one that contains variables and thus does not have a definite truth value until the variables are assigned values. Options a, b, and d all contain variables (x, y) and are therefore open. Option c, "5 is odd", is a statement about a specific constant (5) and has a definite truth value (true), so it is a closed sentence / atomic formula with no free variables.

Explanation

An open predicate is one that contains variables and thus does not have a definite truth value until the variables are assigned values. Options a, b, and d all contain variables (x, y) and are therefore open. Option c, "5 is odd", is a statement about a specific constant (5) and has a definite truth value (true), so it is a closed sentence / atomic formula with no free variables.

4) In P(x): 'x is prime', P is:

A specific number

The predicate or property

The truth value

The domain

In the notation "P(x): x is a prime number", P is the predicate symbol that represents the property of being a prime number. It is not a specific number, not the truth value (which would be true or false for a specific x), and not the domain (which is the set of objects x can be).

Explanation

In the notation "P(x): x is a prime number", P is the predicate symbol that represents the property of being a prime number. It is not a specific number, not the truth value (which would be true or false for a specific x), and not the domain (which is the set of objects x can be).

5) Which requires a logical connective?

X is a student

X loves y

X studies and x passes

X is red

An atomic predicate is one that does not contain any logical connectives (like and, or, not). Options a, b, and d are atomic because they express a single property or relation. Option c contains the word "and", which is a logical connective, so it is a compound statement and requires the connective ∧ to be written formally.

Explanation

An atomic predicate is one that does not contain any logical connectives (like and, or, not). Options a, b, and d are atomic because they express a single property or relation. Option c contains the word "and", which is a logical connective, so it is a compound statement and requires the connective ∧ to be written formally.

6) Which of the following shows an application of a predicate symbol?

X + 2 = 5

P(x)

∀x P(x)

∃x Q(x)

A predicate symbol is a letter that represents a property or relation that can be applied to an argument. In option (b), we are applying P to the argument x. The other options are equations or complete statements, not predicate applications.

Explanation

A predicate symbol is a letter that represents a property or relation that can be applied to an argument. In option (b), we are applying P to the argument x. The other options are equations or complete statements, not predicate applications.

7) What does P(x) represent in predicate logic?

A specific value of x

A property that x may have

A function of x

A constant

In predicate logic, P(x) represents a statement that says the object x has the property P. For example, if P stands for "is even", then P(x) means "x is even". It does not represent a specific value, a function, or a constant.

Explanation

In predicate logic, P(x) represents a statement that says the object x has the property P. For example, if P stands for "is even", then P(x) means "x is even". It does not represent a specific value, a function, or a constant.

8) Which represents “x is greater than 5” using a predicate symbol?

G(x, 5)

X > 5

P(x)

Q(5)

A predicate symbol is a letter (like G) that represents a relation. The statement "x is greater than 5" involves a relation between x and 5, so we use a predicate symbol G with two arguments: G(x,5). Option b uses infix notation but not a predicate symbol explicitly. Option c uses a unary predicate, which would represent a property of x alone, not a relation between x and 5.

Explanation

A predicate symbol is a letter (like G) that represents a relation. The statement "x is greater than 5" involves a relation between x and 5, so we use a predicate symbol G with two arguments: G(x,5). Option b uses infix notation but not a predicate symbol explicitly. Option c uses a unary predicate, which would represent a property of x alone, not a relation between x and 5.

9) Negation of P(x) in direct syntactic form:

¬P(x)

P(¬x)

¬x

P(x) → False

The negation of a statement P(x) is written by placing the negation symbol ¬ in front of it, resulting in ¬P(x). This means "it is not the case that P(x)". The other options are incorrect: P(¬x) applies the predicate to the negation of x, which is not the same as negating the entire statement.

Explanation

The negation of a statement P(x) is written by placing the negation symbol ¬ in front of it, resulting in ¬P(x). This means "it is not the case that P(x)". The other options are incorrect: P(¬x) applies the predicate to the negation of x, which is not the same as negating the entire statement.

10) Which formula is closed due to a quantifier?

P(x) ∧ Q(y)

∃x P(x) ∧ Q(y)

∀x P(x)

P(a)

A closed formula is one that has no free variables. In option c, the universal quantifier ∀x binds the variable x in P(x), so there are no free variables. Option a has free variables x and y. Option b has a quantifier for x but y is still free. Option d has no variables because a is a constant, so it is closed, but not by a quantifier.

Explanation

A closed formula is one that has no free variables. In option c, the universal quantifier ∀x binds the variable x in P(x), so there are no free variables. Option a has free variables x and y. Option b has a quantifier for x but y is still free. Option d has no variables because a is a constant, so it is closed, but not by a quantifier.

11) Express “x is greater than y” using predicate symbol:

X < y

X > y

G(y,x)

G(x,y)

In formal predicate notation, we use a predicate symbol (like G) to represent a relation. The relation "greater than" is typically represented by a predicate that takes two arguments in the order of the relation. So, "x is greater than y" is written as G(x,y). Option c is the opposite relation. Option a and b uses infix notation which is common in mathematics but not the formal predicate notation.

Explanation

In formal predicate notation, we use a predicate symbol (like G) to represent a relation. The relation "greater than" is typically represented by a predicate that takes two arguments in the order of the relation. So, "x is greater than y" is written as G(x,y). Option c is the opposite relation. Option a and b uses infix notation which is common in mathematics but not the formal predicate notation.

12) Domain of P(x): “x is an even integer”:

All numbers

All integers

All real numbers

All primes

The domain is the set of objects that the variable x can take. The property "is an even integer" is defined for integers. While it could be applied to other numbers, the typical domain for this predicate is the set of integers because evenness is a property of integers.

Explanation

The domain is the set of objects that the variable x can take. The property "is an even integer" is defined for integers. While it could be applied to other numbers, the typical domain for this predicate is the set of integers because evenness is a property of integers.

13) Q(x): prime, R(x): x<10, Q(x) ∧ R(x) describes:

All prime numbers

All numbers <10

All primes <10

All evens <10

The symbol ∧ stands for "and". So Q(x) ∧ R(x) means "x is prime and x is less than 10". This describes the set of numbers that are both prime and less than 10. It does not describe all prime numbers (because it requires x<10) nor all numbers less than 10 (because it requires primality).

Explanation

The symbol ∧ stands for "and". So Q(x) ∧ R(x) means "x is prime and x is less than 10". This describes the set of numbers that are both prime and less than 10. It does not describe all prime numbers (because it requires x<10) nor all numbers less than 10 (because it requires primality).

14) N(x) is ¬P(x) where P(x): x>0. N(x) is:

X=0

X is negative

X ≤ 0

X > 0

The negation of "x is positive" (which means x>0) is "x is not positive". This includes two cases: x is negative or x is zero. Therefore, the negation is "x ≤ 0". Option a only covers zero, option b only covers negative numbers, and option d is the original statement.

Explanation

The negation of "x is positive" (which means x>0) is "x is not positive". This includes two cases: x is negative or x is zero. Therefore, the negation is "x ≤ 0". Option a only covers zero, option b only covers negative numbers, and option d is the original statement.

15) Truth value of G(Ada) if Ada studies logic:

True

False

Undefined

Unknown

G(Ada) is a statement about a specific individual, Ada. Since we are told that Ada really studies logic, the statement "Ada studies logic" is true. Therefore, the truth value is True.

Explanation

G(Ada) is a statement about a specific individual, Ada. Since we are told that Ada really studies logic, the statement "Ada studies logic" is true. Therefore, the truth value is True.