Open Sentences and Truth Values

VARIABLES AND TRUTH VALUES

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

A sentence containing a variable is called an \textbf{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.

x+2>7

EXAMPLE

\begin{equation*} x+2>7. \end{equation*}

DEFINITION

1

DEFINITION

p=1

x+2>7

EXAMPLE

x^2=9

SOLVED PROBLEM

7+5=12

SOLUTION

7+5=12

SOLVED PROBLEM

P(x)

SOLUTION

P(x)

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.

Dynamic Sandbox

Initializing Workspace

EXERCISE

Q(x)

ANSWER

x=1

3 Questions

No. An equation with an unassigned variable is usually an open sentence, not a statement.

They give a compact way to record true and false values.

Yes. After assigning a value, the resulting statement may be true or false.

Practise assigning values to open sentences before constructing truth tables for logical connectives.