Predicates

Return to List of Lectures

Previous Lecture

Misc

Has anyone tried the questions and suggestions survey?

Questions?

Predicates and Sets

Parts of section 2.3

Predicates and Open Sentences

Open Sentences. Give some examples of open sentences (not from book), and examples of values that make them true and false

Relevant information from the reading:

• An open sentence is like a statement with variable(s)
• Specific values for the variable(s) make the sentence become a true or false statement

Examples

• x² - 2x = 0. True if x = 2 or x = 0, false if x = 1. So open sentences can take the form of equations.
• x is an integer. True if x = 1, false if x = π. So open sentences can take the form of claims about variables that aren’t equations.
• x² - 10x + 25 = 0. True if x = 5, false if x = 0.
• x is false, or equivalently not x. True if x is the statement “7 = 2,” or if x is the truth value false; false if x = true or x = “2 = 2.” These two different sorts of values for x illustrate why every open sentence needs to also have a universal set specified.
  
  • Universal set for this could be a set of propositions (but beware that if you are going to treat mathematical utterances as mathematical values you get into some subtle issues involving the language the utterance is expressed in and the distinction between the utterance itself and the thing it signifies)
  • Universal set could also be { true, false }, i.e., the set of truth values
• x / 0 = 1. False for all x values because an undefined value is not equal to a number

Truth Sets. What is the truth set for 2x + 3 = 0 if the universal set U is (a) the integers, or (b) the reals

• From reading: The truth set of a predicate is the set of members of the universal set that makes the predicate true
• If U is the set of integers then the truth set for 2x + 3 = 0 is empty
• If U is the set of reals then the truth set = { -1.5 }

Sets

Set builder notation

Form 1 example: { x ∈ ℝ | ...some open sentence using x... } means the set of reals for which the open sentence is true

Problem Set

See the handout

Next

Set builder notation

Next Lecture