Notation for quantified expressions: Be sure to familiarize yourself with the contents of "Appendix C. Notation: quantification" in the document Translating English to Mathematics at http://www.cas.mcmaster.ca/~baber/Courses/General/EnglToMath.pdf#page=32. This appendix defines a notational form for quantification which is more general than most and which, you will find, unifies the several notational forms for quantification you have already encountered in several courses. This notational form for quantification has already been used in many expressions in the MISs presented by the leagues and it will be used on the test and on the final examination in 2B03.
Restricting the domain of a function: Sometimes one wants to define a function that is the same as a given function f except that the new function has a restricted (i.e. "smaller") domain. We write f|X for the function f restricted to the set X. More precisely:
Definition: The function f|X is defined as follows:
f|X(a) = f(a), if aÎX(end of definition)f|X(a) is not defined if aÏX
Therefore, the domain of the function f|X is the intersection of X and the domain of f. More formally,
dom(f|X) = X Ç dom(f)Example: Let the function f be {(1, "a"), (2, "x"), (3, "s")} and the set X be {2, 3, 4}.
The domain of a relation can be restricted in the same way.