Venn diagrams: universal set, empty set and complement

A teacher counts her own class. Eighteen students take French. Fourteen take Spanish. That adds up to thirty two students. There are thirty people in the room. She has not miscounted. Two things are going wrong at once. Some students take both languages, so they got counted twice. Others take neither, so they never got counted at all. One picture separates those two groups.

Start with the picture itself. A box, two circles, and four places a person can be.

Draw a rectangle first. Everything under discussion lives inside it. Here, that is all thirty students in the class. Nothing exists outside the box.

Inside the box, draw two overlapping circles, one per set. Label them F for French and S for Spanish. A student goes inside a circle if they take that language.

That makes four places, and every student sits in exactly one of them. Region one is inside the French circle only. Region two is the overlap, inside both circles. Region three is inside the Spanish circle only. Region four is inside the box, but outside both circles: the students who take neither language. They are still in the class, so they are still in the picture.

Two of those have names you will meet in questions. The overlap is the intersection of F and S. All three regions inside the circles together are the union: everything in one, or the other, or both.

Try one straight away. Sam takes Spanish. Sam does not take French. Which of the four regions is Sam in: one, two, three, or four? Have a think. I'll wait. Sam is in region three. Inside the Spanish circle, outside the French one. Region two would mean Sam takes both. Region four would mean Sam takes neither. Every student is placed by one question, asked twice: in this circle, yes or no.

Now the method. Same class, and this time every region gets a number written in it.

In a survey of thirty students, eighteen study French, fourteen study Spanish, and seven study both. Complete a Venn diagram showing French, Spanish and all thirty students, and find how many study neither. The tempting move is to write eighteen into the French circle. Do not. Eighteen is everyone who studies French, and that already contains the seven who study both. Put eighteen in the French-only region and you have counted those seven twice. So start in the middle. Seven study both, so seven goes straight into the overlap. It is the one number the question hands you outright, and every other region is measured from it. Now work outwards. French has eighteen altogether, and seven of those eighteen are already in the middle. Eighteen take away seven is eleven. Eleven goes in the French-only region. Same move on the other side. Spanish has fourteen altogether, and seven of them are in the middle. Fourteen take away seven is seven. Seven goes in the Spanish-only region. And now the last box: outside both circles. Add up what is inside the circles. Eleven plus seven plus seven is twenty five. The class has thirty. Thirty take away twenty five is five. Five students study neither, and five goes in the space between the circles and the edge of the box. Look at it now. Four regions, four numbers, nothing blank. And the four numbers add back up to thirty, which is your check that the diagram is finished. That settles the register that would not add up. Eighteen plus fourteen came to thirty two because the seven in the middle were counted twice, and the five outside were never counted at all.

Three words for the exam hall: middle first, outside last. Fill the overlap, work outwards into each circle, finish outside both. And underneath the whole topic sits the same question. Which ones do I keep.

Time for the labels. Three symbols live on a diagram like this.

Fresh diagram, small numbers so every element is visible. The numbers under discussion are the whole numbers from two to twelve. Set A is the multiples of five. Set B is the multiples of seven.

Those numbers from two to twelve are the universal set: everything under discussion. On the diagram it is the box itself. Its symbol is a curly capital E, written just inside a corner of the rectangle. Thirteen is not in the universal set here, so thirteen is not on the diagram at all.

Fill it in. Multiples of five from two to twelve: five and ten. So circle A holds five and ten. Multiples of seven: only seven. Five is an element of A, and seven is an element of B. Every remaining number goes inside the box but outside both circles. Two, three, four, six, eight, nine, eleven and twelve. Now every number is somewhere on the diagram, exactly once.

Now look at the overlap. To sit there, a number must be a multiple of five and a multiple of seven. The smallest of those is thirty five, and our numbers stop at twelve. The overlap has nothing in it. A set with nothing in it is called the empty set, and it has its own symbol: a circle with a line struck through it. In the video on sets, definitions and notation, two lists with nothing in common had to be written out in words. This is the symbol that replaces those words. Write it straight into the empty overlap. Be clear what it claims. The empty set is not the number zero, and it is not a gap you left because you were unsure. It says: nothing belongs here, and I have checked.

Last symbol. A dash written after a set name, A dash, is called the complement of A. It means everything that is not in A, while still being inside the box. So shade it in. Take the whole box, then take out the two numbers in circle A, five and ten. What is left is A dash. Two, three, four, six, seven, eight, nine, eleven and twelve. Notice seven is in there. Seven sits inside circle B, but A dash does not care about B. It asks one thing: is this in A. If not, keep it. That is the same question, pointed outwards. For a complement, you keep everything outside.

One more on that same diagram. Which of these is B dash, the complement of B? Option one: five and ten. Option two: every number in the box except seven. Option three: nothing, because B only holds one number.

Have a go. Which one? Option two. B dash is everything inside the box that is not in B: every number from two to twelve except seven. Five and ten are in it, even though they sit inside circle A. Option one is just circle A. Option three swaps the complement for the empty set.

Now, what examiners actually see when they mark these diagrams. First, the counting. An examiners' report on a two-set Venn question, this one about history and geography. If not correct, candidates answered thirteen as they had misunderstood the Venn diagram and not added the five candidates in the intersection representing those studying history and geography. Different subjects, same slip. They read one circle and forgot the overlap belonged to it too. Start in the middle and that mistake has nowhere to happen. Second, the outside. Another board's report gives it one sentence. The most problematic region was the outside, often seen left blank. And the same board again, a year later. Placing values on a Venn diagram was accessible to most students and nearly two thirds gained all three marks. Where one mark was lost, this was usually for omitting the values outside sets A and B but within the universal set, or for a more careless error of missing out one value or misplacing it. Those students did the hard part. They split the totals correctly, then dropped a mark on the one region that needs no thinking. So make it a fixed rule. A two-set Venn diagram is not finished until there is a number in the space between the circles and the edge of the box. If nobody is out there, the number is zero. Write the zero down.

Your turn, with the whole method. Forty students were asked about two clubs. Twenty two go to football club. Sixteen go to drama club. Nine go to both. Put a number in every region, and say how many go to neither club. Take your time. I'll wait right here. Nine goes in the middle first. Football only: twenty two take away nine is thirteen. Drama only: sixteen take away nine is seven. Inside the circles, thirteen plus nine plus seven is twenty nine. Forty take away twenty nine is eleven. Eleven students go to neither club, and eleven goes outside both circles.

That is the lot. Here it is back, in order. The rectangle comes first and holds everything under discussion. Fill the overlap, because each circle's total already includes it. Work outwards into each circle. Then subtract from the total and fill outside both circles. And the three symbols. A curly capital E is the universal set, the box. A circle with a line through it is the empty set, a region with nothing in it. A dash after a set name is the complement: everything outside that set, still inside the box. Which ones do I keep. For a complement, keep everything outside.

Next in the chapter: algebraic set definitions and subsets, on the Higher tier. Sets described by a rule instead of a list, and what it means for one whole set to sit inside another.

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