MATH1011 Chap.1 Sets, Intervals and Set Operations
Sets, Intervals and Set Operations
Choosing between a list, a sentence and a condition
Everything later in this course is phrased in the language of sets, so your first job is to read one collection written three different ways. You can name the members one at a time inside a pair of braces. You can describe them in a sentence precise enough that nobody would argue about who qualifies.
Or you can write a letter, a vertical bar, and the condition that letter has to satisfy. Only the third form keeps working when the collection is infinite, so practise it even where a list would have been quicker.
One habit protects you more than any other: before you read the condition, read what the letter is allowed to be.
The same property asked of the integers and asked of the real numbers gives answers of different kinds, one a short list and the other a stretch of the number line.
Braces record nothing but who belongs, with no order and no tally, so writing a member twice adds nothing and shuffling the entries changes nothing.
Counting members when one member is itself a set
The number of distinct objects a set holds is its cardinality, and this course tests it with a small piece of theatre.
When an inner pair of braces appears in a list, everything between them has been bundled into a single object. So {5, {6, 7}, 8} holds three objects, and the middle one is a collection, neither the number 6 nor the number 7.
The same trap drives the two relations that get mixed up all term. One puts a single object against a collection and asks whether that object is on the list.
The other puts two collections side by side and asks whether the first is wholly reproduced inside the second. So 5 is on the list of that set, while {5} is not, even though everything inside {5} is.
When a mixed list such as {−3, {1}, 0, 6} meets a condition on integers, every braced entry falls out at once: a collection is not an integer whatever is printed inside it.
Building new sets out of two old ones
Four operations do nearly all the work. The union gathers everything appearing in at least one of the two sets. The intersection keeps only what appears in both.
The difference A minus B starts from A and deletes anything B also holds, a different instruction from starting at B, so the order of the letters is not decoration.
The complement is that same deletion measured from the universal set, whatever background the question has declared, so it means nothing until that background is named.
The empty set turns up constantly and is a legitimate answer rather than a sign of error: two sets sharing no member intersect in it. It also sits inside every set, which is why a collection of n objects has 2 to the power n subsets altogether.
Notice that the empty set and the set whose only member is the empty set are different objects: the first holds nothing, the second holds one thing.
From a distance condition to interval notation
Most set questions here arrive wearing absolute value bars, and the quickest reading is geometric: |x − a| is how far x has travelled from a.
Keep that travel under b and x is confined to one stretch of width 2b centred on a. Insist the travel reaches b and x is thrown out of the middle, leaving one piece running left and another running right, reported as a union of two intervals.
Anything written |x + c| is the same picture centred at −c, so turn the plus into a minus rather than guessing where the centre sits.
Each end of your answer carries a bracket inherited from the inequality that produced it: a value the condition allows is written square, one it forbids is written round, and infinity is always round because nothing arrives there.
When two conditions are imposed at once, solve them separately, then combine piece against piece. The surviving piece begins at whichever left end lies further right and stops at whichever right end lies further left, and an end value sitting in both keeps the stricter of the two brackets.
If a bracket is in doubt, put the end value itself back into the original condition and see whether the statement comes out true; that one substitution settles square against round.
What this chapter covers
- 01
Listing, Statement and Rule methods for naming a set
- 02
Cardinality, and why a braced object counts once
- 03
Element or subset: membership against containment
- 04
Union, intersection, complement and difference
- 05
The empty set and the universal set
- 06
Absolute value conditions rewritten as intervals
Worked example · free
Two distance conditions, one intersection, and a list with braces in it
- 2Unfold the strict condition on P into a double inequality and read off the interval.
- 3Unfold the condition on Q into two half lines, then intersect with P one piece at a time and list the integers that survive.
- 2Test each member of M against P, treating every braced object as a single object that is not a number.
Key terms
- Listing Method
- The Listing Method names a set by writing out every one of its members, separated by commas, inside a single pair of braces. It suits small finite collections and becomes unusable the moment the collection is infinite.
- Proper Subset
- A proper subset is contained in a set without exhausting it: every one of its members belongs to the larger set, and the larger set keeps at least one member back. The set itself is the only subset that fails this test, so a set of 4 members has 16 subsets and 15 proper ones.
- Universal Set
- The Universal Set is the background collection a question declares, holding every object that question is willing to discuss. Complements are measured from it, so the complement symbol carries no meaning until this background has been named.
- Set Difference
- The Set Difference A minus B keeps the members of A that do not also belong to B. Reversing the two letters asks a different question and normally produces a different collection.
- Empty Set
- The Empty Set holds no members whatever, and is written either as a bare pair of braces or with the slashed circle symbol. It lies inside every set, it is the honest answer whenever two sets share nothing, and it is a different object from the set whose only member is the empty set.
Sets, Intervals and Set Operations FAQ
Does the set {2, {3, 4}, 5} have three members or four?
Three. The inner braces wrap 3 and 4 into one object, so the set holds the number 2, a two-member collection, and the number 5. That inner collection is neither the number 3 nor the number 4, which is why an intersection with a set of integers throws it away.
What changes if a set is built from integers rather than real numbers?
The condition stays the same and the answer changes shape completely. Over the real numbers a distance condition gives you intervals, reported in bracket notation. Over the integers the very same condition gives a finite list you can write out and count, which is why questions asking how many members there are almost always name the integers first.
Is A minus B the same as B minus A?
Almost never. A minus B keeps the part of A that B does not reach, while B minus A keeps the part of B that A does not reach, and those two collections share nothing. They agree only in the single case where both come out empty, which happens exactly when the two sets hold identical members.
Exam move
Take ten absolute value conditions off the tutorial sheets, half of them strict and half allowing equality, and write only the interval for each with no working shown, until the bracket at each end arrives without thought. Then invent one mixed list of eight objects with three of them braced, and count how many belong to a stated interval and how many survive a stated integer condition.
Those two drills, done in that order, cover most of what this chapter is actually tested on.
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