What Is the Truth Value of P(x,y,z)?

The predicate $P(x,y,z)$ is defined by the condition $x+y\le z$. That definition tells you how to test any particular values of $x$, $y$, and $z$. However, the question as provided does not include the statement whose truth value you are meant to find. Without those values or a complete quantified statement, there is no single truth value to report.

What the predicate means

A predicate is a condition that can be true or false once its variables have values. Here, calculate the sum of the first two values and compare it with the third:

$$P(x,y,z)\text{ is true exactly when }x+y\le z.$$

If $x+y$ is greater than $z$, then $P(x,y,z)$ is false. Equality counts as true because the definition uses $\le$, not just $<$.

For instance, $P(2,3,5)$ is true: $2+3=5$, and $5\le5$. In contrast, $P(2,3,4)$ is false because $2+3=5$ and $5\nleq4$.

How to evaluate a complete statement

First identify what is being asked about the predicate:

• Specific values: Substitute the values for $x$, $y$, and $z$, then check the inequality.
• A quantified statement: Use the quantifier and the stated domain. A universal claim must hold for every allowed choice; an existential claim needs just one allowed choice that works.
• A compound statement: Evaluate the predicate first, then apply the logical connectives or negation in the rest of the statement.

For a quantified example, suppose the variables range over the real numbers. The statement $\exists x\exists y\exists z\,P(x,y,z)$ is true: choosing $x=1$, $y=2$, and $z=3$ gives $1+2\le3$. By contrast, $\forall x\forall y\forall z\,P(x,y,z)$ is false. The choice $x=1$, $y=1$, and $z=1$ makes the condition $2\le1$, which is false.

The domain matters because quantifiers range over that domain. In the examples above, the same witness and counterexample work if the domain is the integers. For a different domain, check that the values used are allowed.

If the intended relation is strict less-than

If the definition was meant to be $P(x,y,z): x+y<z$, equality no longer makes the predicate true. For example, with $x=2$, $y=3$, and $z=5$, the non-strict version is true, but the strict version is false because $2+3$ is not less than $5$.

When $x+y<z$, the predicate is true only if the third value is strictly larger than the sum. When $x+y\le z$, it is true if the third value is either larger than or equal to the sum. Check the symbol in the actual statement before deciding.

Common mistakes

• Treating the predicate definition as a complete proposition: $P(x,y,z)$ still depends on the variables. You need values or quantifiers to settle its truth value.
• Reversing the comparison: The condition compares $x+y$ with $z$; it is not $x+y\ge z$.
• Ignoring equality: For $\le$, equality makes the predicate true. For $<$, it makes it false.
• Assuming a domain: Quantifiers such as “for every $x$” need a domain, such as the integers or real numbers, to specify what values are allowed.

What information is needed for the specific answer?

To determine the requested truth value, provide the omitted expression—for example, a particular instance such as $P(2,3,5)$, or a quantified statement such as $\forall x\exists z\,P(x,1,z)$—and specify the domain if it is not already clear. Then substitute the values or test the quantifiers using the definition of $P$.