Sets: a bag of things, without duplicates

A set is a bag of things. You can throw Mila, Tom and Ana into one bag and call it A. The bag does not care about order, and it does not care how many times you dropped Mila in: a set holds each thing once. If you ask "is Tom in the bag?", the only possible answers are true or false. There is no "kind of".

Three operations cover most of what you will ever need. The union (A ∪ B) is the bag you get when you tip both bags into one: everything from A, everything from B, each name once. The intersection (A ∩ B) is only what sits in both bags. The difference (A \ B) is what is in A but not in B. Union reads as "or", intersection reads as "and", difference reads as "not in".

A = { Mila, Tom, Ana }
B = { Tom, Ana, Ben }

A ∪ B  (union, "or")          = { Mila, Tom, Ana, Ben }
A ∩ B  (intersection, "and")  = { Tom, Ana }
A \ B  (difference, "not in") = { Mila }

  Tom ∈ A   -> true
  Ben ∈ A   -> false

Membership is a yes/no question:

This matters because a set answers a question with a yes or a no, and computers are extremely good at yes-or-no questions. Every "have I seen this user before?" and every "does this word already exist?" is a membership test, and the whole point of picking a set over a list is that the set answers much faster.