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Real intervals and sets of real numbers

STACK has a simple system for representing and dealing with real intervals and sets of real numbers.

Simple real intervals may be represented by the inert functions oo(a,b), oc(a,b), co(a,b), and cc(a,b). Here the character o stands for open end point, and c for a closed end point. So oc(-1,3) is the interval \( \{ x\in\mathbb{R} | -1 < x \mbox{ and } x\leq 3.\} \), and is displayed as \( (-1,3] \) with mismatching brackets in the tradition of UK mathematics.

The Maxima function union requires its arguments to be sets, and intervals are not sets. You must use the %union function (from the package to_poly_solve) to join simple intervals and combine them with discrete sets. E.g. %union(oo(-2,-1),oo(1,2))

Note that the %union function sorts its arguments (unless you have simp:false), and sort puts simple intervals of the form oo(-inf,a) out of order at the right hand end. So, some sorting functions return lists of intervals, not %union as you might expect, to preserve the order.

As arguments, the %union command can take both simple intervals and sets of discrete real numbers, e.g.

%union(oo(-inf,0),{1},oo(2,3));

Similarly, STACK provides %intersection to represent an intersection of intervals (which the package to_poly_solve does not have).

Predicate functions

  1. intervalp(ex) returns true if ex is a single simple interval. Does not check ex is variable free, so oo(a,b) is a simple interval. {}, none, all and singleton sets are not considered "intervals" by this predicate, use realsetp instead. The primary purpose of this predicate is to detect intervals oo, oc etc within code.
  2. inintervalp(x, I) returns true if x is an element of I and false otherwise. x must be a real number. I must be a set of numbers or a simple interval of the form oo(a,b) etc.
  3. trivialintervalp(ex) returns true if ex is a trivial interval such as \((a,a)\).
  4. unionp(ex) is the operator a union?
  5. intersectionp(ex) is the operator an intersection?
  6. realsetp(ex) return true if ex represents a definite set of real numbers, e.g. a union of intervals. All end points and set elements must be real numbers, so oo(a,b) is not a realset. If you want to permit variables in sets and as endpoints use realset_soft_p instead.
  7. interval_disjointp(I1, I2) establishes if two simple intervals are disjoint.
  8. interval_subsetp(S1, S2) is the real set S1 contained within the real set S2?
  9. interval_containsp(I1, S2) is the simple interval I1 an explicit sub-interval within the real set S2? No proper subsets here, but this is useful for checking which intervals a student has.

Basic manipulation of intervals.

  1. interval_simple_union(I1, I2) join two simple intervals.
  2. interval_sort(I) takes a list of intervals and sorts them into ascending order by their left hand ends. Returns a list.
  3. interval_connect(S) Given a %union of intervals, checks whether any intervals are connected, and if so, joins them up and returns the ammended union.
  4. interval_tidy(S) Given a union of sets, returns the "canonical form" of this union.
  5. interval_intersect(S1, S2) intersect two two simple intervals or two real sets, e.g. %union sets.
  6. interval_intersect_list(ex) intersect a list of real sets.
  7. interval_complement(ex) take a %union of intervals and return its complement.
  8. interval_set_complement(ex) Take a set of real numbers, and return the %union of intervals not containing these numbers.
  9. interval_count_components(ex) Take a set of real numbers, and return the number of separate connected components in the whole expression. Simple intervals count as one, and sets count as number number of distinct points in the set. Trivial intervals, such as the empty set, count for 0. No simplification is done, so you might need to use interval_tidy(ex) first if you don't want to count just the representation.

Natural domains, and real sets with a variable.

The function natural_domain(ex) returns the natural domain of a function represented by the expression ex, in the form of the inert function realset. For example natural_domain(1/x) gives

realset(x,%union(oo(0,inf),oo(−inf,0)));

The inert function realset allows a variable to be passed with a set of numbers. This is mostly for displaying natural domains in a sensible way. For example, where the complement of the intervals is a discrete set, the realset is displayed as \(x\not\in \cdots\) rather than \(x \in \cdots\) which is normally much easier to read and understand.

realset(x,%union(oo(0,inf),oo(-inf,0)));

is displayed as \(x \not\in\{0\}\).

Validation of students' answers

Students must simply type union (not %union) etc.

Validation of students' answer has a very loose sense of "type". When we are checking the "type" of answer, if the teacher's answer is a "set" then the student's answer should also be a "set" (see setp). If the teacher's answer is acually a set in the context where an interval should be considered valid, then the teacher's answer should be the inert function %union, e.g. %union({1,2,3}), to bump the type of the teacher's answer away from set and into realset.

Validation does some simple checks, so that mal-formed intervals such as oo(1) and oo(4,3) are rejected as invalid.

Assessment of students' answers

The algebraic equivalence answer test will apply interval_tidy as needed and compare the results. Currently the feedback in this situation provided by this answer test is minimal.


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