Predicates Quiz: Understand Functions, Relations, and Logic Structure

A predicate contains variables and has no truth value until those variables are assigned. A predicate such as P(x) does not assert a complete claim because x is unspecified. Until a value is substituted for x, the statement cannot be judged as true or false, making it an open statement.

Predicates with variables are open statements and require instantiation to evaluate. An open statement depends on one or more free variables, so it remains without a truth value. Only after replacing each variable with a specific constant does the expression become a proposition.

Once x is replaced by 7, the statement becomes truth-evaluatable, so it is a proposition. Instantiating x removes the free variable, turning the expression into a closed form that can be evaluated as true or false. Propositions contain no free variables.

A predicate with two arguments, such as R(x, y), has arity 2. Its truth value depends on the pair (x, y), so it is called a binary or two-place predicate. Since 5

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Any real number squared is non-negative, making the statement universally true. If the predicate asserts x² ≥ 0 over the real numbers, every possible substitution satisfies the condition, because squaring a real number never produces a negative value.

D(x,y) represents the relation "x is a divisor of y". A relation with two inputs can describe number-theoretic concepts. Here, D(a, b) is true exactly when a divides b evenly, so the predicate expresses divisibility.

To instantiate a predicate, substitute the constant directly into the parentheses. Turning P(x) into P(4) simply replaces the variable with a constant. This removes the open nature of the statement and allows a truth evaluation.

The presence of a variable makes the statement open, lacking a truth value until assignment. An open sentence such as Q(x) is incomplete, because without a specific x, it is unclear what claim is being made. Only after substitution does it become a definite statement.

Earth replaces the variable x, functioning as a constant argument. By inserting Earth into H(x), the variable disappears and the resulting sentence, such as “Earth is a planet,” can be checked for truth without needing further information.

P(7) has no free variables and can be evaluated as true or false. Once the substitution is made, the statement no longer depends on any unassigned variable and expresses a definite claim whose truth can be determined.

Predicates with subjective properties depend on contextual definitions to determine truth. Properties like “tall” or “large” have meaning only after defining the relevant scale or domain. The predicate’s truth varies based on context rather than pure logical structure.

Both 2 and −2 satisfy the equation, so they make the predicate true. If the predicate states x² = 4, substituting either 2 or −2 produces a correct equality. These are the exact values that make the statement true.

¬R(x) means “x is not even,” but that does not guarantee that x is “odd” unless the domain is restricted to the integers.

S(x) means “x is positive.”

Thus the expression means “x is positive and not even,” which is not logically the same as “x is positive and odd” in general domains.

A predicate with two arguments is a binary predicate. Two arguments indicate that the expression defines a relation between ordered pairs of objects, and evaluating it requires specifying both inputs.

Containing a variable, it is not truth-evaluatable until instantiated. Any unassigned variable leaves the statement incomplete. The truth value cannot be determined until every variable is replaced by a constant.

Substituting a constant eliminates free variables, producing a proposition. The act of replacing the variable closes the statement. The resulting form is no longer open and can be tested for truth logically or computationally.

Arity denotes the number of arguments a predicate accepts. A unary predicate has arity 1, a binary predicate has arity 2, and so on. The arity determines how many inputs the predicate requires for evaluation.

Ground atomic formulas contain no variables—only constants and a predicate symbol. An expression like R(a, b) is completely instantiated and can immediately be evaluated because all variables have been replaced with specific constants.

Only y is bound by the quantifier; x and z remain free. In a statement such as ∃y P(x, y, z), the quantifier applies only to y. The variables x and z still appear without any binding quantifier, so the overall statement’s truth depends on their unspecified values.