Higher Mathematics · Class 9-10 · Chapter 1 · Sets and functions
Set operations, and Venn diagrams.
So far sets have been asked about; now they get put to work. There are four ways to make a new set out of two old ones, and each of the four has a picture. Once the picture is in your hand the symbols stop being something to memorise and become something to look at.
1. Four operations, four words
The union, written A ∪ B, means take anything that is in either one. The “or” here is inclusive: being in both is no objection. For example {2, 4, 6, 8} ∪ {3, 6, 9} = {2, 3, 4, 6, 8, 9}. Notice that 6 is in both sets and yet appears once in the answer, because repeats are not counted in a set.
The intersection, A ∩ B, means it has to be in both. In the example above the intersection is {6}, that being the one number in both sets. And the difference, A \ B, means in A but not in B. In the same example {2, 4, 6, 8} \ {3, 6, 9} = {2, 4, 8}, because the 6 has dropped out.
The fourth is the complement, A′, which is U \ A: in the universal set but not in A. Here is where the point from the last section earns its keep. Until U has been named there is no saying where “outside” is, so without U a complement means nothing. Tap the eight symbols below and see what each gives when run over the same three sets.
symbolstap a symbol
U = {1…6}, A = {1, 2, 3, 4}, B = {3, 4, 5}
Union. Take anything in either, so it is {1, 2, 3, 4, 5}. The 3 and the 4 are in both, and still written once.
Intersection. It has to be in both, so it is {3, 4}.
Difference. In A but not in B, so it is {1, 2}.
The difference the other way. It is {5}. Nothing in common with A \ B at all, because difference does not commute.
Complement. In U but not in A, so it is {5, 6}. Without knowing U this could not have been worked out at all.
The complement of the union. A ∪ B = {1, 2, 3, 4, 5}, so its complement is {6}. Only the 6 is left outside both circles.
Intersection with a complement. B′ = {1, 2, 6}, so A ∩ B′ = {1, 2}, which is exactly A \ B. The difference and this intersection are one and the same.
The empty set. It is ∅. Nothing can be inside A and outside A at once.
All eight seen. Three things worth keeping: difference does not commute, A \ B = A ∩ B′, and A ∩ A′ is always empty.
Put the third and fourth of that list side by side. A \ B = {1, 2} and B \ A = {5}, with not one member in common. Swapping the two sets changes nothing for union and intersection, but it changes everything for difference. Drag the handle below and watch the shaded part move on one unchanging picture.
before and afterdrag the handle
A \ B, or B \ A?
A \ B
B \ A
2. All of them on one picture
A Venn diagram is a simple thing: the universal set is a rectangle, and inside it each set is a circle. Two circles cut the rectangle into four regions: A only, B only, the overlap, and outside both. Forgetting that last one is the commonest mistake, and it is the very region a complement needs.
Now the real point. Every one of these operations is the answer to one question: which regions do I take? Press the buttons and watch the picture stay put while only the shaded regions change.
regionspress a button to light a region
Every operation on one picture
A. A only, plus the overlap: those two regions together are all of A. A = {1, 2, 3, 4}.
B. B only, plus the overlap. B = {3, 4, 5}.
A ∪ B. Take all three regions and leave only the outside. It is {1, 2, 3, 4, 5}.
A ∩ B. Only the sliver in the middle. It is {3, 4}.
A \ B. The part of A that did not get into B, that is A with the overlap taken off. It is {1, 2}.
A′. Everything outside A but inside U. So not only the B only region but the outer corner as well. It is {5, 6}.
All six seen. The picture never changed; only which regions were taken changed. That is all an operation is: a choice of regions.
check yourself
One question
With U = {1, 2, 3, 4, 5, 6, 7}, A = {1, 2, 3, 4} and B = {3, 4, 5}, what is A′ ∩ B?
Right. First A′ = U \ A = {5, 6, 7}, then intersect with B = {3, 4, 5}, and only the 5 is in both. Notice this equals B \ A as well.
That is A ∩ B, with the complement not taken. A′ means outside A, so 3 and 4 drop out.
That is A′ on its own, with the intersection with B still to do. Neither 6 nor 7 is in B.
That is A \ B, the difference the other way. What was asked for is the part of B that is not in A.
3. Three circles, eight regions
With three sets, how many regions? You can count them rather than remember them. One yes or no question per circle: am I in A? in B? in C? Three questions give 2 to the 3, that is eight answers, so eight regions. It is the same count as the power set, with inside or outside in place of take it or leave it.
The widget below runs the other way from every other Venn diagram. Usually you are given an expression and you shade the picture; here you shade the picture and the name comes out. In an exam it is this direction that is the real skill. Tap regions to shade them and read off what you have made.
you picktap a region
The eight region game
Two things are worth noticing. First, some choices have more than one name, and then all of them are shown joined by equals signs. Those are set identities, and it is how the two De Morgan laws turn themselves up. Second, some choices have no short name at all; that is not a failure, it is still a perfectly good set, only there is no brief way to write it.
Switch modes and take the De Morgan challenge. There are two of them, and between them they say one thing twice: taking a complement turns the symbol over.
check yourself
Another question
In a three circle Venn diagram, inside the universal set, how many separate regions are there?
The three pairwise overlaps and the middle have not all been counted. One yes or no per circle gives 2 to the 3.
The outside region has been forgotten. “In none of the three” is a region like any other, and it lies inside the universal set.
Right. One yes or no question per circle, so three questions give 2 to the 3 = 8 answers, and eight regions. The outside is one of them.
One too many. There cannot be more than eight, because three yes or no answers give every point its full address.
One last question, easiest if you picture it. Disjoint sets are ones with A ∩ B = ∅, which in a Venn diagram means the two circles do not even touch.
check yourself
Last question
If A and B are disjoint, which of these must be true?
It is the intersection that is empty, not the union. Two separate circles can be full; nothing is simply in both.
Right. Taking B's members out of A finds nothing to take, because not one member of A is in B. So A comes through untouched.
That would hold only if A and B together covered the whole of U. Being disjoint does not do that; there can be room outside both circles.
Rather the opposite. The region outside both circles is precisely A′ ∩ B′, and it is not usually empty.
The lines from this note that the next section needs for proving identities are cut onto the cards below.
cut from the notetap to turn over
Four cards
frontAre A \ B and A ∩ B′ the same?
backYes, exactly. “In A but not in B” and “in A and outside B” say the same thing.
frontHow many regions with three circles, and why?
backEight. Inside or outside for each circle, so three questions give 2 to the 3.
frontWhat does (A ∪ B)′ equal?
backA′ ∩ B′. Take a complement and a union turns into an intersection.
frontWhat must you know before finding a complement?
backThe universal set U. A′ = U \ A, so without U the word outside means nothing.