Sets: definitions and notation (union, intersection, element of)

Two of you, one pizza, one topping list to agree on. You want mushroom, pepperoni and olives. Your mate wants olives, sweetcorn and pepperoni. Order every topping either of you named and someone spends the meal picking things off. Order only the toppings you both named and everyone eats. Two different jobs, done on the same two lists. One combines them. The other keeps only the overlap. Maths gives each job its own symbol, and the two symbols look almost identical. One is a cup. One is a cap. Point the wrong one at a question and you answer the opposite question.

Start with the word itself. What a set actually is, and how you write one down.

A set is a collection of things, written inside curly brackets with commas between them, and given a capital letter for a name. Maya's hobbies: M equals football, tennis, chess. The things inside are called elements. Football is an element of M. Two rules follow, and both come from one idea. A set only records which things are in it. Not how many times, and not in what order. So chess, tennis, football is exactly the same set as football, tennis, chess. And you never write the same element twice.

That gives us the first symbol. It is a small rounded E, and it means is an element of. Football is an element of M. It asks a yes or no question about one single thing: is that in this set, or not.

Try one straight away. Maya's set, M, is football, tennis, chess. Three statements. A: painting is an element of M. B: tennis is an element of M. C: netball is an element of M. Exactly one of them is true. Which one? Have a think. Which one is true? It is B. Tennis is an element of M, because tennis is written inside those curly brackets. Painting and netball are not in there, so both of those statements are false. If you got that, you can already use the symbol.

Now the two symbols from the pizza. The cup, and the cap.

The cup is called union. A union B means every element that is in A, or in B, or in both, all gathered into one set. It is the or symbol. Pour both lists into the cup, and the cup catches the lot.

The cap is called intersection. A intersection B means only the elements that are in A and in B. It is the and symbol. The cap sits over both lists and keeps only what is underneath both of them.

So carry this into the exam hall. Cup catches everything. Cap keeps only the shared. And underneath both, one question: which ones do I keep. Every symbol in this topic is an instruction about which elements you keep.

Maya and Leo. Maya's hobbies, M, are football, tennis and chess. Leo's hobbies, L, are chess, painting and football. Write down M intersection L, and M union L. Intersection first, so the cap: keep only the shared. Chess is on both lists. Football is on both lists. Tennis is only Maya's. Painting is only Leo's. Neither of those survives. So M intersection L is football and chess. Now the cup: keep everything. Football, tennis, chess, painting. That is M union L, and it is every hobby either of them does. Count that union though. Six hobbies were written down between the two of them, but the union has four elements, not six. Chess and football were on both lists, and a set never repeats an element, so each one goes in once. Repeats collapse.

Sets do not always arrive as a list. Sometimes you get a rule, and you build the list yourself.

Here is one. A equals, open curly bracket, x, colon, x is a factor of twelve, close curly bracket. That colon is read as such that. So: A is the set of all values of x, such that x is a factor of twelve. A rule, not a list.

Turn the rule into a list. Factors of twelve, taken in pairs: one times twelve, two times six, three times four. So A is one, two, three, four, six and twelve. And B equals x such that x is a positive even number less than ten. That list is two, four, six and eight. Ten itself is not less than ten, so ten stays out. Now the element symbol earns its keep. Is six an element of A. Yes, six is on that list. Is six an element of B. Yes again, six is even and under ten. Six is in both sets. So write down A intersection B. Every number that appears on both lists. A is one, two, three, four, six, twelve. B is two, four, six, eight. Take your time. I'll wait right here. A intersection B is two, four and six. Those three are on both lists. Eight is even, but eight is not a factor of twelve. Twelve is a factor of twelve, but twelve is not less than ten. Each one fails half the test, so neither gets in.

One more pair of lists. This pair does something the first two did not.

Amir's paint colours, X, are red, blue and green. Priya's paint colours, Y, are yellow and purple. Write down X intersection Y. Go through Amir's colours one at a time. Red is not on Priya's list. Blue is not on Priya's list. Green is not on Priya's list. So X intersection Y has no elements in it at all. Amir and Priya share no paint colours. That is a real, correct answer, and for now you write it in words: X and Y have no elements in common. There is a symbol for a set with nothing in it. It arrives in the video on Venn diagrams, where the universal set, the empty set and the complement all get their symbols. Until then, words do the job.

Time to look at all this the way an examiner does. Examiners see that nothing-in-common case go wrong, and not because the maths is hard. One examiners' report on this topic puts it like this: Beyond this, there were many muddled, ambiguous and wrong statements and numerous blanks whilst some did not recognise the empty set symbol. Read what that actually says. Blanks. Muddled statements. Those are students who worked out the right thing and then could not write it down. Working it out and recording it are two separate skills, and it is the second one that gets marked. The other one to watch is the cup and the cap. Before you write anything, look at the symbol in the question and say it in your head. Cup, so everything. Cap, so only the shared. And check the size of your answer: a union is at least as long as the longer list, an intersection is never longer than the shorter one.

Your turn, with numbers this time. P is two, three, five and seven. Q is one, three, five, seven and nine. Write down P intersection Q. Then write down P union Q. Then decide whether nine is an element of P. Pause the video and work all three out. I'll wait. P intersection Q is three, five and seven: the numbers sitting on both lists. P union Q is one, two, three, five, seven and nine. Six elements, not nine, because the three shared numbers each go in once. And nine is an element of P? No. Nine is on Q's list only, so that statement is false.

Everything in this video comes down to three symbols. The small rounded E means is an element of: one thing, one yes or no. The cup means union: everything from both sets, each element written once. The cap means intersection: only what the two sets share. And when they share nothing, you say that in words.

Next in the chapter: Venn diagrams, the universal set, the empty set and the complement. Those same two symbols get a picture there, two overlapping circles, and the set with nothing in it finally gets a symbol of its own.

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