After studying implication and biconditional, it is important to return to a subtle question: when does a sentence containing a variable become a statement? Many mathematical sentences contain variables, but their truth values are not fixed until values are assigned. This distinction separates open sentences from statements. In this lesson, open sentences and truth values will connect algebraic conditions with logical statements. :::definition[Open Sentence] A sentence containing a variable is called an open sentence if its truth value depends on the value assigned to the variable. An open sentence becomes a statement after a value is assigned to the variable. ::: An open sentence may look like a mathematical statement, but it is incomplete for truth-value purposes. The expression $x+2>7$ is not true or false until $x$ is assigned a value. This does not make the sentence meaningless. It means that the sentence is waiting for a value from a specified context or domain. :::example[Open Sentence] Consider the sentence $$\begin{equation*} x+2>7. \end{equation*}$$ This sentence is not a statement until a value is assigned to $x$. If $x=6$, then $$\begin{equation*} 6+2>7 \end{equation*}$$ is true. If $x=3$, then $$\begin{equation*} 3+2>7 \end{equation*}$$ is false. Thus, the truth value depends on the assigned value of $x$. ::: :::definition[Truth Value] The truth value of a statement is either true or false. In truth tables, the value $1$ denotes true, and the value $0$ denotes false. ::: :::definition[Truth Assignment] A truth assignment assigns a truth value to each primitive statement under discussion. For example, assigning $p=1$ and $q=0$ means that $p$ is true and $q$ is false. ::: There are two common ways to make an open sentence into a statement. First, assign a specific value to the variable. For example, $x+2>7$ becomes a statement after taking $x=6$. Second, later in logic one may use quantifiers such as for all or there exists, but quantifiers are not needed for the present truth-table discussion. For this section, the central idea is that truth tables require definite truth values. :::example[Assigned Truth Values] Let the open sentence be $x^2=9$. If $x=3$, then the sentence becomes $3^2=9$, which is true. If $x=4$, then the sentence becomes $4^2=9$, which is false. Hence, the same open sentence can produce different truth values under different assignments. ::: :::solved-problem Determine whether each expression is a statement or an open sentence: (1) $7+5=12$. (2) $x-1=4$. (3) $n$ is divisible by $3$. (4) $21$ is divisible by $3$. (5) $y^2\geq 0$ for a specified real number $y$. ::: :::solution [1] The expression $7+5=12$ has a definite truth value. It is a statement. [2] The expression $x-1=4$ depends on the value of $x$. It is an open sentence. [3] The expression $n$ is divisible by $3$ depends on the value of $n$. It is an open sentence. [4] The expression $21$ is divisible by $3$ has a definite truth value. It is a statement. [5] If a specific real number $y$ is already assigned, then $y^2\geq 0$ has a definite truth value. It is a statement after the assignment. ::: :::solved-problem Let $P(x)$ be the open sentence $x+2>7$. Find the truth value of $P(2)$, $P(5)$, and $P(8)$. ::: :::solution Let $P(x)$ denote $x+2>7$. For $x=2$, $$\begin{align*} 2+2&>7\\ 4&>7, \end{align*}$$ which is false. Hence, $P(2)=0$. For $x=5$, $$\begin{align*} 5+2&>7\\ 7&>7, \end{align*}$$ which is false. Hence, $P(5)=0$. For $x=8$, $$\begin{align*} 8+2&>7\\ 10&>7, \end{align*}$$ which is true. Hence, $P(8)=1$. ::: Use the checker to turn an open sentence into a statement by assigning a value. Choose an expression pattern, enter a number, and evaluate the resulting sentence. The output shows the substituted statement and its truth value. This reinforces the idea that an open sentence is not true or false until the variable receives a value from the chosen domain. :::calculator[Open Sentence Truth Value Checker] ::: :::exercise Let $Q(x)$ be the open sentence $2x-3>5$. Determine the truth values of $Q(1)$, $Q(4)$, and $Q(6)$. ::: :::answer For $x=1$, $2(1)-3=-1$, so $Q(1)=0$. For $x=4$, $2(4)-3=5$, so $Q(4)=0$ because $5>5$ is false. For $x=6$, $2(6)-3=9$, so $Q(6)=1$. ::: :::faq[Frequently Asked Questions] Q: Is every equation a statement? A: No. An equation with an unassigned variable is usually an open sentence, not a statement. Q: Why do truth tables use $1$ and $0$? A: They give a compact way to record true and false values. Q: Can an open sentence become false after assignment? A: Yes. After assigning a value, the resulting statement may be true or false. ::: :::call-to-action[Prepare for Truth Tables] subtitle: Practise assigning values to open sentences before constructing truth tables for logical connectives. button: Next Lesson | /discrete-mathematics/mathematical-logic/basic-logical-connectives-and-truth-tables/truth-tables-for-basic-connectives button-ghost: Previous Lesson | /discrete-mathematics/mathematical-logic/basic-logical-connectives-and-truth-tables/implication-and-biconditional :::