Lesson 1 in Perceptive Mathematics: the concept of a set

 Look at this photo of eight apples.

How many apples are there in the photo? Eight. How many photos are there? One. Several objects or entities seen as one thing is called a set. The objects that make up the set are called its elements. Unlike in a photo, the arrangement of these objects among each other, or any information about the elements of the set apart from knowing what these elements are, is not considered to be part of the information about the set. Thus, for instance, the set of the eight apples that we see in the photo would be the same as the set of the same apples shown in a different photo, where these apples have been rearranged.

If A1, A2, ..., A8 are symbolic representations of the apples shown in the photo, then {A1, A2, A3, A4, A5, A6, A7, A8} is how we symbolically represent the set of those apples. 

Exploration 1.1. According to your interpretation of the definition of a set given above, which of the following sets should be the same set as the set {A1, A2, A3, A4, A5, A6, A7, A8}?
  • {A1, A3, A2, A4, A5, A6, A7, A8}
  • {A1, A2, A3, A4, A5, A6, A7}
  • {A1, A2, A3, A4, A4, A6, A7, A8}
  • {A1, A2, A3, A4, A4, A5, A6, A7, A8}
Just like we could add apples to a photo, or take some away, we can add elements to a set, or take them away. For instance, let us start with the set {A1, A2, A3}. If we take A3 away from this set, we get the set {A1, A2}. What happens if we take away A1 and A2 as well? Can we do that? Well, taking away A2 gives us {A1}, and taking away A1 from this set should perhaps give us {}? Such a set, that is, a set having no elements, is also allowed to be a set. It is called the empty set.

Exploration 1.2. Start with a set of three apples and a set of three mountains (represent the mountains by M1, M2, M3). Take away one element from each set, and do this three times. Will the resulting sets be the same set or will they be different sets? More generally, is there only one empty set, or are there many empty sets?