Mathematical Statements

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Questions?

Problem sets and Canvas?

• I’ll post problem sets through Canvas.
• However, Canvas isn’t a terribly good match for assignments with face-to-face grading, so what it will say about due dates is only part of what matters (e.g., it says problem set 1 is due at 11:59 next Monday; the reality is that you need to finish it by Monday, but 11:59 is just the latest time I could put on that day, and after you finish doing the work you still need to come show it to me before 5:00 PM Thursday).
• I also want to give you grades in person (via the face-to-face meetings), and review them with you in person, rather than leaving you to figure out what they mean through Canvas, so at least for this semester Canvas will not have information on your grades in this course.

Statements

Section 1.1

Is “Prof. Baldwin owns two cats” a mathematical statement?

• Statement = declarative sentence that’s either true or false
• This example meets those requirements, so it is a mathematical statement

Which of the following are statements, and are the ones that are statements true or false?

• If -1 is a natural number, then Prof. Baldwin is an aardvark
  • This is a true statement, by the rule that conditionals with false hypotheses are true.
• If x is a real number, then x² ≥ 0
• If x > 7, then x² ≥ 0
  • The above two examples demonstrate the problematic nature of variables in statements.
  • The problem is that a variable can stand for many different values, and so you generally can’t determine whether a sentence with a variable in it is true or false
  • But some phrases define a set of possible values for a variable to represent, and knowing that set sometimes lets you determine whether the sentence as a whole is true or false; in such cases, the sentence is a statement.
    • For example, “if x is a real number, then x² ≥ 0” can be read as stating that all real numbers have squares greater than or equal to 0, which is a true statement
    • In contrast, “if x > 7, then x² ≥ 0” does not provide information about a set of values that x can represent (what if x is imaginary or otherwise not comparable to 7), and so isn’t a statement.
• For all real numbers x, x² ≥ 1
• For some real number x, x² ≥ 1
  • The above two examples are both statements, and demonstrate that the phrases “for all” and “for some” are very important in determining whether a statement is true or false (the first is false, but the second is true).
• If Fido is a dog, then Fido has fur
  • In some sense, “Fido” here is just a variable, and all the previous discussion of variables applies to it.
    • i.e., this example can be read as the statement “all dogs have fur,” which is false because some dogs are furless.
  • But another problematic thing about variables is that context is important in determining what a variable can represent — if this sentence occurs in a context where “Fido” names a specific thing, e.g., “I hear you have a pet named Fido; if Fido is a dog ...” then the truth or falsehood of the conditional can be determined based on whether Fido is or isn’t a dog and does or doesn’t have fur. Similarly in a context such as “let x be a real number; if x > 7 ...” the truth or falsehood of a conditional that doesn’t by itself define x can be determined.
• This sentence is false
  • This is not a statement because it is paradoxical, i.e., it can’t be either true or false.
  • If the sentence is true, i.e., it is true that the sentence is false, then the sentence is false.
  • On the other hand, if the sentence is false, then clearly it is true that “this sentence is false.”

Problem set

• Finish it by Monday, but turn it in by meeting with me sometime before 5:00 Thursday
• Make appointments for meetings via Google calendar (I’ll demonstrate how to do this in class on Monday)

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Proofs

Read section 1.2

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