Predicates, Free Variables, and Atomic Formulas Quiz

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| Questions: 15 | Updated: Dec 1, 2025
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1) Which of the following is a predicate rather than a proposition?

Explanation

A predicate is a statement that contains  variables and becomes a proposition only when the  variables are replaced by specific  values. Option B contains the  variable x, so it is a predicate. The other options are either commands (A) or statements without  variables (C and D), which are propositions because they have a definite truth  value.
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About This Quiz
Predicates, Free Variables, And Atomic Formulas Quiz - Quiz

Think you can spot the difference between a predicate with free variables and a closed proposition? This quiz focuses on the structure of predicate formulas: free vs. bound variables, well-formed formulas, atomic predicates, and ground atomic formulas. You’ll analyze expressions like P(x), P(a), ∀x P(x), and S(x, y) to decide... see morewhich are open, which are closed, and which are just single predicate applications versus compound formulas. Along the way, you’ll deepen your understanding of n-ary predicates and how instantiating variables with constants turns a schema into a specific statement. By the end, you’ll have a clear mental picture of how predicate logic sentences are built. see less

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2) What makes a predicate different from a proposition?

Explanation

A predicate is a statement that involves a predicate symbol (like P, Q) applied to arguments, and these arguments can include  variables. A proposition, on the other hand, is a statement that is either true or false and does not contain  variables. While predicates can be used with quantifiers (A) or logical connectives (D), these are not the defining features.
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3) Which is a well-formed formula containing a free variable?

Explanation

A well-formed formula is a syntactically correct expression. A free  variable is a  variable that is not bound by a quantifier. In option A, x is a free  variable because there is no quantifier for x, and Q is a proposition (no  variables). Option B is not well-formed because y and z are  variables without a predicate.
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4) P(3) is a:

Explanation

P(3) is a statement where the  variable in the predicate P has been replaced by a constant (3). Therefore, it has no free  variables and is a closed formula, which is also called a proposition. It is not a predicate with a free  variable (B), not a quantified statement (C), and not a schema (D) because it is a specific instance.
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5) Which has a free variable?

Explanation

A free  variable is one that is not bound by a quantifier. In option D, x is a free  variable because there is no quantifier for x. In options A and C, the  variables x and y are bound by quantifiers. In option B, a is a constant, so there are no  variables.
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6) Which is the simplest atomic predicate (no function symbols)?

Explanation

An atomic predicate is one that does not contain logical connectives or quantifiers. Option D is simply a predicate symbol applied to a  variable, which is the simplest form. Options A and B contain logical connectives (∨ and ∧), so they are compound. Option C contains a function symbol f, which is not allowed in the simplest atomic predicate.
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7) Translated to normal English language, 'Student(x)' means:

Explanation

The expression "Student(x)" is a predicate that says that the object x has the property of being a student. It does not make an existential claim (B) or a universal claim (C), nor does it say that x is not a student (D).
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8) Which of the following is an atomic predicate?

Explanation

An atomic predicate is a single predicate symbol applied to terms, without logical connectives and no quantifiers. Option C is just a predicate symbol applied to two arguments, so it is atomic.
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9) Which of the following is an open formula rather than a sentence?

Explanation

An open formula contains free  variables and is not a complete sentence (proposition). Option A has free  variables x, so it is an open formula. The other options are sentences: B and D are quantified (no free  variables), and C uses a constant.
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10) Which is true about P(x) and P(2)?

Explanation

P(x) contains a free  variable x, so it is a predicate (not a proposition). P(2) is the result of replacing x with the constant 2, so it has no free  variables and is a proposition. Therefore, option B is correct.

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11) Which of the following is not a single predicate symbol application?

Explanation

A predicate symbol application is a single predicate applied to arguments, such as P(a,b), Q(x), or S(z). Option C is a compound formula that uses the logical connective ∧ to combine two predicate applications, so it is not a single predicate symbol application.
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12) Which best describes an n-ary predicate?

Explanation

An n-ary predicate is one that takes n arguments. For example, a unary predicate takes one argument, a binary predicate takes two, and so on. It is not defined by the number of quantifiers (A), the number of  variables (which might be more than the arity if some are bound), or depth (D).
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13) Which is a ground atomic formula?

Explanation

A ground atomic formula is an atomic predicate that contains no  variables. Option B uses a constant a, so it is ground. Options A and C contain  variables, and option D is quantified and not atomic.
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14) The predicate R(x,y) has an arity of:

Explanation

The arity of a predicate is the number of arguments it takes. R(x,y) takes two arguments, x and y, so its arity is 2. This is fixed and does not depend on the domain.
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15) Identify the free variables in P(x,z) ∧ ∃y Q(y):

Explanation

A free  variable is one that is not bound by a quantifier. In the expression, x and z are not bound by any quantifier, so they are free. The  variable y is bound by the existential quantifier ∃y, so it is not free.

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Which of the following is a predicate rather than a proposition?
What makes a predicate different from a proposition?
Which is a well-formed formula containing a free variable?
P(3) is a:
Which has a free variable?
Which is the simplest atomic predicate (no function symbols)?
Translated to normal English language, 'Student(x)' means:
Which of the following is an atomic predicate?
Which of the following is an open formula rather than a sentence?
Which is true about P(x) and P(2)?
Which of the following is not a single predicate symbol application?
Which best describes an n-ary predicate?
Which is a ground atomic formula?
The predicate R(x,y) has an arity of:
Identify the free variables in P(x,z) ∧ ∃y Q(y):
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