Hi. My name is Dianna and although i know that your suppost to only go up as high as high school on this site, i just have a basic algebra question. Im a freshman in college, but its really basic. I want to know why its okay to say that, for example, 6 is an element of the set of integers, but you get counted off for saying that the set of 6 is an element of the set of integers. How come? Thank you for your time.

Hi Dianna,

Here's an anology that I like to draw when I teach set theory. I think of a set of items as a box with those items in it. For example, I might visualize the set with the integers 1,2,3,...,10, as a cardboard box containing the plastic numbers 1,2,3,...,10. Then if I look in the box, I can find the element 6, namely a plastic 6. However, in my analogy, the set 6 would be a cardboard box containing a plastic 6. If I look inside my box with the numbers 1,2,3,..., 10, I can't find a little cardboard box with a 6 in it. Thus the element 6 is in my big box of numbers, but the set 6 is not.

The box anology also works well when trying to understand the emptyset or sets of sets. The empty set can be viewed as an empty cardboard box. It isn't "nothing", but rather a box which contains nothing. In your class you may talk about the set containing the emptyset. This would be a box containing an empty box, which is clearly different from an empty box. The set with the integers 1,2,3,...,10 in it is written {1,2,3,...,10}. The set containing the set 1, the set 2, the set 3,..., the set 10, written {{1},{2},{3},...,{10}}, would be a box containing ten little boxes, with each little box containing one of the numbers 1,2,3,...,10. Now the element 6 is not in this big box (it's in a little box inside the big box), but the set 6 (a little box with a 6 in it) is in the big box. Thus, the set 6 is an element of the set {{1},{2},{3},...,{10}} but not an element of the set {1,2,3,...,10}. Tricky, I know, but hopefully this analogy will help you keep it straight. Everytime you see the word "set", put the things that follow into a cardbox box in your imagination.

Hope this helps,

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