Download Understanding Quantifiers: Universal and Existential in Mathematics - Prof. Margaret M. Fl and more Study notes Discrete Structures and Graph Theory in PDF only on Docsity! Crash Introduction to Quantifiers Margaret M. Fleck 30 January 2009 This lecture finishes the quick introduction to quantifiers, which follows part of section 1.3 of Rosen. We’ll also see some examples of direct proof, from section 1.6 of Rosen. Don’t worry about reading either section right now. We’ll do that when we’re closer to covering their full content. 1 Announcements Turn in your homework: please put it in the folder for your section on the back table. Announcement from Fernanda Mendes about the UIUC WCS (“Women in Computer Science”) mentoring program. Match you up with corporate mentors. Open to everyone (not just women). Details on their web site. 2 Recap Recall from last lecture that a predicate (e.g. x2 ≥ 0) is a statement that becomes true or false when you substitute particular values for its variables (e.g. 32 ≥ 0). Predicates are used to make general claims about whole sets of values, such as “For every integer x, x2 ≥ 0.” Notice that we carefully specified the type of x, i.e. what types of values (integers in this case) that we allow to be substituted for the variable x. The set of values that can be substituted in for a variable is called its domain or its replacement set. 3 Quantifiers The operation “for all x” is formally known as a universal quantifier. 1 The other heavily-used quantifier in formal mathematics is the existential quantifier, written “there exists an x” For example, “there exists an integer x such that 5 < x < 100.” This means that at least one integer x (and possibly a whole bunch of integers) satisfies the equation. Notice that the existential quantifer is followed by “such that” in fluent mathematical English, which isn’t quite parallel to what you see with the universal quantifier. This isn’t for any good reason. It’s just how math- ematical English happens to have evolved, and you should do likewise so that your written mathematics looks professional. “Such that” is sometimes abbreviated “s.t.” The general idea of a quantifier is that it expresses express how many of the values in the domain make the claim true. Normal English has a wide range of quantifiers which tell you roughly how many of the values work, e.g. “some”, “a couple”, “a few”, “many”, “most.” Mathematics largely uses just the two we’ve seen, though you occasionally see a third quantifier “there exists a unique x.” This means that one and only one x satisfies the predicate. For example, “there is a unique integer x such that x2 = 0.” Mathematicians use the adjective “unique” to mean that there’s only one such object (similar to the normal usage but not quite the same). In a statement like ∀x ∈ N, x ≤ 2x, the quantifier ∀ is said to “bind” its variable x. Similarly, summations and integrals bind their dummy vari- ables, e.g. the i in ∑ n i=0 1 i . Variables that aren’t bound are called “free.” Statements containing free variables don’t have a defined truth value. 4 Shorthand notation The universal quantifier has the shorthand notation ∀. For example, ∀x ∈ N, x ≤ 2x. The existential quantifier is written ∃, e.g. ∃y ∈ Ry = √ 2. Notice that we don’t write “such that” when the quantifier is in shorthand. The unique existence quantifier is written ∃! as in ∃!x ∈ R, x2 = 0. There’s several conventions about inserting commas after the quantifier and/or parentheses around the following predicate. The style in these notes and the style in Rosen are both ok to copy from. If you want to state a claim about two numbers, you can use two quan- tifiers as in: ∀x ∈ R, ∀y ∈ R, x + y ≥ x 2
