Mathematical Statements

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Anything You Want to Talk About?

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Mathematical Statements

…Also known as “propositions.”

Based on the first part of section 1.1 in the textbook, and the statements discussion.

What Is a Statement?

Which of the examples in the Canvas discussion are mathematical statements?

1. * It sometimes snows in Geneseo. Whether this is a statement depends on what “sometimes” means to you. If you have some informal bound on how often something needs to happen to count as “sometimes” (e.g., if it last snowed in Geneseo 30 years ago, would you nonetheless say that it “sometimes” snows?) then it’s not a statement, because different people’s informal bounds can lead to different beliefs about whether it’s true or false. On the other hand, if “sometimes” literally means that there are or were some times when it snowed in Geneseo, then this is a statement, and it’s true.
2. It snowed in Geneseo on February 1, 2021. This is a statement.
3. SUNY Geneseo is the best college. This is not a statement because it’s a matter of opinion.
4. * It rained in Geneseo on June 7, 1849. This is a statement, even if no-one here (or maybe anywhere) knows whether it’s true or false. Since we can all agree that it had to have either rained or not rained in Geneseo on that date, the assertion is either true or false and therefore a statement.
5. Go Knights! Exclamations are not statements.
6. There is some integer x such that 3x + 2 = 11. This is a statement.
7. * To solve the equation 3x + 2 = 11, subtract 2 from both sides and then divide by 3. This is a command not a statement. (But if you read it as “One way to solve the equation is....” then it becomes a statement about a valid way to use algebra.)
8. Is x = 3 a solution to 3x + 2 = 11? Questions aren’t statements.
9. x = 3 is a solution to 3x + 2 = 11. This is a statement.
10. x = 4 is a solution to 3x + 2 = 11. This is also a statement, but it’s false.
11. * x = 3 is a solution to 3x + 2 = 11 and so is x = 4. This is a statement, but seeing why involves thinking deeply about the “and.” Basically, it says that there are 2 solutions, roughly “x = 3 and x = 4 are both solutions,” which is not true. Therefore it’s an assertion that is unambiguously false, and thus a statement. More formally, “and” requires both parts to be true.
12. Read a book. This is another command, not a statement.
13. * This statement is false. This is neither true nor false, it’s a paradox. (If it’s true that “this statement is false,” then the statement must be false, not true. On the other hand, if the statement is false, then the claim “this statement is false” is actually true.)

Can you think of some examples of mathematical statements of your own?

• There is an integer x such that x + 1 = 2.
• sin 90° = 1.
• George Washington was born on a Tuesday. A nice example of a claim that isn’t about a mathematical subject, and we don’t know right off if it’s true or false, but it is one or the other and so it meets the definition of “mathematical statement.”

How to Tell If a Statement is True

What could you do to determine if these statements from the discussion are true or false? At least in some cases, what could you do to prove it (i.e., persuade everyone else that the statement is true/false)?

1. There is some integer x such that 3x + 2 = 11. True, because x = 3 satisfies 3x + 2 = 11
2. For all integers x, 3x + 2 = 11. False because there’s an example that falsifies it, e.g., x = 4.
3. For every integer n > 2, there are no integers a, b, and c all greater than 0 such that aⁿ + bⁿ = cⁿ. Fermat’s Last Theorem.
4. For every integer n > 4, every n^th-degree polynomial has at least one non-integer root.
5. Every even integer greater than 2 is the sum of two prime numbers. The Goldbach Conjecture.
6. At noon on February 6, 2021, atmospheric pressure in Geneseo will have been between 14.68 and 14.72 pounds per square inch.

The first 2 of these demonstrate what you can and can’t do with examples. Specifically, a claim that some thing exists can be proven true by showing an example that makes it true, and a claim about all things can be proven false by showing an example (a “counterexample”) of a thing for which the claim doesn’t hold.

Items 3 and 5 are famous conjectures that went unproven for a long time after they were discovered. Fermat’s Last Theorem went unproven for around 300 years, until it was proven in the early 1990s, and the Goldbach Conjecture is still unproven. Both demonstrate that simply looking for examples or counterexamples usually doesn’t work, and therefore why more powerful forms of proof are necessary. (Item 4 looks like it might fall in the same category, but if you think about it you can construct counterexamples — see the Canvas discussion for one.)

Item 6, atmospheric pressure, is another example of a statement that isn’t about a mathematical subject, and so the method of proof wouldn’t be particularly mathematical: you could just go measure the atmospheric pressure at that time.

Problem Set

Problem set 1, on mathematical statements, is now available.

You should have it finished by next Friday (February 12), and graded by the following Friday (February 19).

Next

“Conditional” statements, i.e., statements of the form “if something is true, then this other thing is true.”

Particularly common and important in mathematics, partly because many theorems take this form, and partly because this form allows you to make deductions.

Please read “Beginning Activity 2 (Conditional Statements)” and “Conditional Statements” in section 1.1 of the textbook.

Please also participate in this discussion of conditional statements by class time Monday.

Remember: Monday we are face-to-face for Cohort A, in Welles 140.

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