Three-set Venn diagrams, element counts and n(A) notation (Higher)

Three sign-up sheets on a desk. Drama, coding, chess. Forty students in the year group, and the year head needs one number by lunchtime. How many signed up for nothing at all. That number is on none of the sheets, and you cannot count your way to it. Plenty of students are on two sheets, a few on all three, so reading straight down the lists counts them twice, or three times over. One picture sorts it out. It has eight places a name can land.

One flag before the maths starts. This is Higher tier content. If you are sitting Foundation, this one is not on your exam, so feel free to skip it: the video on Venn diagrams, the universal set, the empty set and the complement covers what you need. On Higher, stay put. Two skills here, marked separately.

First, the picture. Three circles, and why they make eight places rather than six.

Same idea as the two-circle diagram, drawn one size up. A box holds all forty students. Inside it, three circles, one per club, each overlapping the other two, and all three overlapping in the centre. Eight places, and here is where eight comes from. Each student answers three yes or no questions. Drama, yes or no. Coding, yes or no. Chess, yes or no. Two answers, three times over. Two times two times two is eight, and every combination gets a region. Name them out loud. The centre, for all three clubs. Three lens shapes for exactly two clubs. Three regions for one club only. And one inside the box, outside every circle. Every one of those eight ends up with a number in it, a written zero included where nobody belongs, exactly as in the video on Venn diagrams, the universal set, the empty set and the complement. But note what this picture is for. All three circles cut across each other because the job is counting regions. It is not the one-set-inside-another drawing from the video on algebraic set definitions and subsets.

Try the move this whole method runs on. Six students do drama and coding. Two of those six do all three clubs. So how many do drama and coding, but not chess? A: six. B: four. C: two. Take a moment. Which one? B, four. That six counts everybody in the drama and coding overlap, and two of them carry on into the centre. Take those two out and four are left. If you got that, you have done the step the rest of the method is built from.

One question has run under every symbol in this topic. Which ones do I keep. A three-circle diagram is that question answered eight times at once. Keep drama, drop coding, drop chess. Keep all three. Keep none.

Now the method in full, on those sign-up sheets from the desk.

Forty students were asked about three after-school clubs: drama, coding and chess. Fifteen do drama. Eighteen do coding. Twelve do chess. Six do drama and coding. Five do coding and chess. Four do drama and chess. Two do all three. Find how many do none of them. Start in the centre, and here is the reason. Every other number you were handed is contaminated. Six do drama and coding, and that six already contains anyone doing all three. Fifteen do drama, and that fifteen covers four separate regions. The two is the only figure meaning exactly what it says. So two goes straight into the centre region. Drama, coding and chess, all three. Now the three lens shapes, one subtraction each. Six do drama and coding, take away the two in the centre, leaves four for drama and coding only. Five do coding and chess, take away the same two, leaves three. Four do drama and chess, take away two, leaves two. Every one of those strips out the same thing: the centre, which each pair total has already counted. Say the subtraction out loud as you write it. Six take away two, because the two are further in. That sentence is the step. Now the single-club regions. Each takes off everything already inside its own circle. Drama holds fifteen, and inside drama sit four, two and two. Fifteen, take away four, two and two, leaves seven doing drama only. Coding holds eighteen, and inside it sit four, three and two, which leaves nine doing coding only. Chess holds twelve, and inside it sit two, three and two, leaving five doing chess only. Seven regions done. Add them. Two, four, three, two, seven, nine and five make thirty two students placed inside the circles. The year group is forty. Forty take away thirty two is eight. Eight do none of the three clubs, and eight goes outside all three circles. That is the number that was on none of the sheets. And you can prove the diagram is finished: eight regions, eight numbers, adding back up to forty.

Second skill now, and a genuinely separate one. Same finished diagram, one new piece of notation.

Two students are handed the same diagram and asked for n of A. One writes two. The other writes curly brackets, four, seven. Both have read the picture correctly. Only one has answered the question asked. It is written as a lower case n, then A inside round brackets, and it means one thing. The number of elements in A. Not the elements themselves. How many of them there are. n is for number. Two instructions now sit side by side, looking almost identical. Write down A: that answer is a list, in curly brackets. Write down n of A: that answer is a single number, no brackets, no commas. One asks which ones. The other asks how many. That is the line to carry into the exam hall. n asks how many, not which ones. Back to the clubs. Write down n of Drama. Not the drama-only region: n of Drama is the whole drama circle, all four of its regions. Seven, four, two and two. That adds to fifteen, the exact fifteen the question started with. It checks itself. Your go, same diagram. Write down n of, open bracket, Coding, then the cap-shaped symbol, read as intersection, then Chess, close bracket. Is it A: three. B: five. Or C: eight? One number. Take your time. B, five. The intersection of coding and chess is not just the lens shape. It is everything those two circles share, and the centre is shared too. Three in the lens plus two in the centre is five. Option A stopped at the lens. And notice the answer's shape. One number. One more off the same picture. n of none of the three is eight. The same eight you worked out, now asked for by name.

Now the exam corner, and what examiners report on each of those two skills. The diagram first. An examiners' report on a Higher paper, on a question about knitting and photography. The Venn diagram was challenging to a fair number who did not understand what they needed to do, for instance to take away 9 from the 17 who chose knitting and photography. Read what that actually says. The arithmetic was not the problem; seventeen take away nine is not hard. They did not know which number came off which. That is what centre-first buys you: the order tells you what each subtraction is for. Now the notation. This next report names parts of a question you cannot see, so listen past the letters, to what the students wrote. In (c) and (d), some misinterpreted the meaning of n in brackets, viewing the values in the Venn diagram as individual elements rather than the number of elements. Consequently, they provided answers such as 2, for 2 elements, or curly brackets 4, 7 in part (b), and similarly 3 or curly brackets 8, 9, 11 in part (d). Look at what they wrote. A two. A three. Two little lists in curly brackets. Every one of those answers a question nobody asked: they read values off the diagram, which is the which-ones question. So two checks before you write an n answer down. One: are you counting the whole region the notation names, centre included. Two: is it a single number. A comma or a curly bracket means you answered which ones instead.

Your turn, both skills at once. Thirty students were asked about three sports. Fourteen play football. Twelve play netball. Eleven do athletics. Five play football and netball. Four do netball and athletics. Three do football and athletics. One does all three. Put a number in every region. Then write down n of football, and how many do none of the three. Take your time. I'll wait right here. One goes in the centre. Then the lenses: five take away one is four, four take away one is three, three take away one is two. Then the singles. Football: fourteen, less four, two and one, leaves seven. Netball: twelve, less four, three and one, leaves four. Athletics: eleven, less three, two and one, leaves five. Inside the circles that is twenty six students. Thirty take away twenty six is four, and four goes outside all three circles. And n of football is fourteen. The whole circle, exactly as the question handed it to you.

Two separate skills went past there. Worth taking them back out one at a time. Building it. Centre first, because it is the only number meaning exactly what it says. Then the lens regions: each pair total, take away the centre. Then the single regions: each circle total, take away everything already inside. Then outside: the total, take away everyone placed. Reading it. n of something is a count, and a count is one number. n of a circle means the whole circle, every region in it. n of an intersection means everything the two circles share, centre included. And that closes the thread this chapter has run on. Which ones do I keep. It sits under every symbol here, and a finished three-circle diagram answers it eight times over. n is the one piece of notation asking something else. How many.

That completes our chapter on sets and Venn diagrams. Sets written down and the notation for them. Two-circle diagrams with the box, the empty set and the complement. Sets defined by a rule, and one set inside another. And now three circles, eight regions and counting with n. The next chapter is rounding, estimation and bounds.

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