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Writing and Classifying True, False and Open Statements in Math

Instructor: Michael Quist

Michael has taught college-level mathematics and sociology; high school math, history, science, and speech/drama; and has a doctorate in education.

In math you need to be able to know whether a statement is true, false, or open. In this lesson, you'll learn how to classify and write true, false, or open statements.

What Are True, False, and Open Statements?

You probably know what a lie detector does. A person is connected up to a machine with special sensors to tell if the person is lying. As math students, we could use a lie detector when we're looking at math problems! It would make taking tests and doing homework a lot easier! In this lesson, we'll look at how to tell if a statement is true or false (without a lie detector). We'll also look at statements that are open, which means that they are conditional and could be either true or false.

A true statement is one that is correct, either in all cases or at least in the sample case. For example, the number three is always equal to three. It is also equal to six divided by two. Any variable, like x, is always equal to itself. Also, if you have two quantities that are equal and you perform the same operation on both quantities, you'll end up with another set of equal quantities - another true statement.

A false statement is one that is not correct. For example, the number three is not equal to four, so a statement that says that three and four are equal would be false. Three is not equal to six divided by three, so 3 = 6/3 would also be a false statement. Or if you add unequal quantities to opposite sides of a true equation, you'll end up with a false statement.

An open statement is one that may or may not be correct, depending on some unknown. For example, you could be asked if x = 3. Well, it's true if the value for x in that problem happens to be 3. Otherwise, it's false. Until you establish a real value for x, the statement is considered open. It depends on facts that you don't have.

Classifying True, False, and Open Statements

A statement is true if it's accurate for the situation. A true statement does not depend on an unknown. In mathematics, we use rules and proofs to maintain the assurance that a given statement is true. If you start with a statement that's true and use rules to maintain that integrity, then you end up with a statement that's also true. Proofs are the mathematical courts of truth, the methods by which we can make sure that a statement continues to be true.

In math, statements are generally true if one or more of the following conditions apply:

  • a math rule says it's true (for example, the Reflexive Property says that a = a)
  • a math problem gives it as an initial condition (for example, the problem says that Tommy has three oranges)
  • the statement can be reached through a logical set of steps that start with a known true statement (a proof)

Alternatively, a false statement is one that is not accurate for the situation at hand. In math, a statement is false if one or more of the following conditions apply:

  • it contradicts a math rule (for example, if you say that aa or a > a)
  • it contradicts a piece of information given in the math problem (for example, if the problem says that Tommy has three oranges and you write down four oranges instead)
  • it contradicts a set of logical steps that start with a known true statement (for example, if you know that two quantities are equal and then you say that they are equal after adding different amounts to each side)

An open statement is one where you don't have enough information (or have not found it yet) to determine whether it's true or not. You can tell open statements in math by looking for conditions.

  • There is a stated condition or question in the problem (for example, 'Does Mary earn more than $10 per hour?')
  • There are variables in the statement that could make it true under certain conditions (for example, x = y if they both happen to be equal to the same value)

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