We've just learned about predicates, which are propositions whose truth depends on the value of one or more variables within the predicate. Predicates are different from propositions in that they can sometimes be true but not always. It's helpful to be able to speak of when a predicate is true -- either sometimes or always. This process is called quantification. In this unit we will learn about existential quantification and universal quantification
Basic objectives: Each student is responsible for gaining proficiency with each of these tasks prior to engaging in class discussions, through the use of the learning resources (below) and through the working of exercises (also below).
- Identify a quantifier as either existential or universal and use appropriate notation.
- Write a quantified statement given in English using symbols.
- Write a quantified statement given in symbols using English.
- Determine whether a quantified statement is true or false.
Advanced objectives: The following objectives are the subject of class discussion and further work; they should be mastered by each student during and following class discussions.
- Form the negation of a quantified statement (either universal or existential).
- Interpret a predicate with two quantifiers and determine whether the statement is true or false.
To gain proficiency in the learning objectives, use the following resources. You may include other resources if you wish, in addition to or in replacement of the following.
Textbook: In Applied Discrete Structures, read Section 3.8. Make sure to read actively, working through examples and activities as you go.
Video: Watch the following videos. Note that the video uses yet another term, "open sentence", which is also a synonym for "proposition over a universe". That is, "proposition over a universe" = "predicate" = "open sentence".
- Quantified statements (10:05)
- Negating quantified statements (10:00)
The following exercises are to be done during and following your reading and viewing of the resources. Work these out on paper and then enter the responses into the appropriate submission form (see Submission Instructions) by the deadline. You will receive a mark of Pass if each item response shows a good-faith effort to be right and is submitted prior to the deadline.
- Suppose that the universal set
$U$ is the set of all GVSU professors and$P(x)$ is the predicate "Professor x has a Ph.D. degree." Consider the statement,$(\forall x \in U)(P(x))$ . Explain why this quantified statement is a proposition now, and not a predicate. - Again consider the statement,
$(\forall x \in U)(P(x))$ using the same data as in question 1. Write out what this sentence says, using clear English. - Suppose the universal set
$U$ is now$\mathbb{Z}$ , the set of all integers. Consider the predicate$Q(n)$ which says, "n! < 2^n". Now consider the quantified statement$(\exists n \in \mathbb{Z})(Q(n))$ . Is this statement true, or is the statement false? There are two items at the submission form, one to indicate your answer and another to explain your reasoning. - Using the same universe and predicate as in question 3, consider the quantified statement
$(\forall n \in \mathbb{Z})(Q(n))$ . Is this statement true, or is the statement false? There are two items at the submission form, one to indicate your answer and another to explain your reasoning.
Submit your responses using the form at this link: http://bit.ly/1OggtcE