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## Statistics and probability

### Course: Statistics and probability>Unit 7

Lesson 3: Basic set operations

# Subset, strict subset, and superset

A subset of a set A is any set B such that every element of B is also an element of A. A strict subset is a subset that isn't equal to the original set (i.e. B must have at least one fewer element than A). A superset of A is any set C such that A is a subset of C. Created by Sal Khan.

## Want to join the conversation?

• What's the difference between a 'subset' and a 'strict subset'? If there is a strict subset than a 'not-strict' subset should exist as well. Could you please give an example of such difference? •   Like he mentions at , the difference is that a strict subset cannot be the same set, that is, it cannot contain all of the elements that the other set does. Or in other words, a strict subset must be smaller, while a subset can be the same size.

As an example, if A = {4,7} and B = {7,4} then A is a subset of B (because B contains all of the elements A does), but A is not a strict subset of B, because both sets contain exactly the same elements (B does not contain any element that A does not have).
However, if C = {7,4,1} then A is both a subset of C, and a strict subset of C, because C contains all the numbers A does and also a number A does not contain (the number 1).
• So subsets and supersets are the same thing? •  No, subsets have all their elements in another set while supersets contain all the elements of another set, though they may have more elements. Think of it this way...
A { 1,2,3,4 } B { 2,4 } B is a subset of A because all of its elements exist in A, A is not a subset of B because not all of its elements exist in B, but A is a superset because it contains all the elements of B and B is not a superset of A because B doesn't contain all of the elements of A, just 2.
• Is it necessary to have the underline? if not, is the "equal to" property assumed without the underline? • Unfortunately, different mathematicians define these symbols in slightly different ways. Some say A⊂B to mean that A is a subset of B and A⊊B to mean that A is a proper subset of B. Other mathematicians say A⊆B to mean that A is a subset of B and A⊂B to mean that A is a proper subset of B. In theory, some mathematicians could also do like Sal and say A⊆B to mean that A is a subset of B and A⊊B to mean that A is a proper subset of B, although I don't actually recall ever seeing that in a textbook or paper. Mostly, it depends on the style of the mathematician and whether subsets or proper subsets are the more important concept in whatever branch of math you're reading about.
• is a null set a strict subset of itself • No. For two sets A and B, the proper subset relation A ⊂ B implies that B contains at least one element which is not contained within A.

Denoting the null set with ∅, the statement A ⊂ ∅ would imply that ∅ contains at least one element which is not in A. However, the null set contains no elements, so the statement is impossible. There cannot be a proper subset of ∅. The null set is a subset of itself, but not a proper (strict) subset of itself.
• if A ϵ B, B ⊂ C, then why not A ⊂ C ? • When would you use sets in real life? • There are a few answers, but one answer is that more advanced math needs a rigorous foundation and often uses theorems and results that depend on set theory.

Advanced math can be so abstract or complicated that you must make sure your foundations are logically sound with no room for guess work or intuitive definitions.

The set of real numbers which applied maths uses everyday is constructed from the union of the sets of natural, rational, irrational and transcendental numbers.

One way to start this whole construction is starting with the construction of the natural numbers. One way to construct these natural numbers is using set theory! More information about this construction can be found at: http://en.wikipedia.org/wiki/Natural_number#A_standard_construction

So set theory allows you construct the foundations of mathematics and use advanced math that needs a rigorously constructed foundation.
• I was taught that the symbol for “is a member of,” or what we called “is an element of,” ("E" for "Element") is the lowercase Greek letter epsilon, or ε. Sal says that it is not. I’m OK with that. But here is something that I just noticed:

1. The "subset" symbol is a “C-shaped” symbol with a line below it.
2. The “element of” symbol is a “C-shaped” symbol with a line through it.
3. Continuing that same logic, if we keep the line rising, we would get a “C-shaped” symbol with a line above it.

Here’s my question:
Does such a symbol exist, and if it does, what does it mean?
I might have gone with “superset,” but apparently, superset is represented by the subset symbol written backwards.
(I can’t find a way to type the symbol I want, so I may have actually answered my own question.) But please think this through with me anyway. • You're not totally wrong on the set membership notation. I am given to understand that it was originally a lowercase epsilon. But when set theory became the foundation for all of modern mathematics, a Greek letter was too important a symbol to give up. So the set membership symbol was tinkered with so that you could say "Let ε∈ℙ be given" without any fear of misunderstanding.

You're also right that there seems to be no symbol in Unicode for a subset symbol with a bar over it. If you ever wrote a paper that used that symbol it would mean whatever you defined it to mean, at least in that paper.
• So, isn't every set a subset and a "strict" subset of the Universal Set? • Say there are 2 sets- A and B.
A is {2, 5, 8}
and B is {2, 5, 8}
What are all the relationships?
Are A and B just subsets of each other?

Thanks!
(1 vote) • A = B, so:
A would be a subset of B,
B would be a subset of A,
A would be a superset of B,
and B would be a superset of A.
However,
A cannot be a strict subset of B,
B cannot be a strict subset of A,
A cannot be a strict superset of B,
and B cannot be a strict superset of A.
Hopefully this helps you. Tell me if I didn't properly answer your question, or if you still need help. :) 