Overlapping Sets

Start with some collection of objects and break up the set into two parts, where one part is a majority and the other is a minority, like this.

The red circles form a majority since there are 5 total items, and the red ones account for 3 (more than half) of them. Now try to get a majority with only the remaining black elements. You can’t! That means that any other majority that you try to make by including (even all of) the black elements will also need to include a red element.

Now let’s talk about a real-life use of this fact. Let’s say you have a group of people that are planning a party, and they’re voting on certain aspects of the party (like when to have it and what food to serve, … Pick your favorite aspects of a party). For this example, let’s say you’re voting on whether the party should be on a Friday or Saturday, and whether the food served should be pizza or hot dogs. After the vote, suppose that more than half of the group voted to have the party on Saturday and more than half of the group voted to serve pizza.

Then you know that there must have been at least one person who voted to serve pizza at a party on Saturday! If you vote on more than two decisions, each with two options, then for any pair of winning items, there must have been at least one person who voted for both of them!