Discrete Math: Subsets and Venn Diagrams Explanation

In summary, A is an element of B but not a subset of B, while {A} and {{A}} are both elements and subsets of B. The venn diagram approach is not suitable for this problem, and the key to understanding is that "A is a subset of B if and only if every element in A is in B".
  • #1
carlodelmundo
133
0

Homework Statement



Let their be a set A, and let B be the set: {A, {A}} (the set containing the elements A and the
set that contains element A)

As you know, A is an element of B and {A} is also an element of B.

Also, {A} is a subset of B and {{A}} is also a subset of B.

However, A is not a subset of B

Homework Equations



[URL]http://65.98.41.146/~carlodm/phys/123.png[/URL]


The Attempt at a Solution



See my drawing above. I created a Venn Diagram to deduce the logic with no clear results. In the first diagram to the left, I can clearly see that the element A is a subset of B ... yet.. it is not? Can someone explain?
 
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  • #2
I don't think Venn diagrams are the way to go. Let's do an example. Let A be the set of integers. We know that B has two elements: A and {A}. But A has 3 as an element. 3 is neither the set of integers or the set containing the set of integers, so 3 is not in B.

In the venn diagram you are getting A as an object and A as a set containing other objects confused
 
  • #3
My question:

Why are we comparing set A (the set of all integers) on an element-by-element basis? I agree that "3" is an element of A such that A = { ...,-1,0,1,0,1,2,3...}.

I agree that "3" is neither the set of integers A = { ...,-1,0,1,0,1,2,3...} nor the set containing the set of integers {A} = {{ ...,-1,0,1,0,1,2,3...}} .

I fail to see how individual elements of A are pertinent to the problem. If A is a subset of B, every element of A is in B.

----------------------------------------------------- Light Bulb in My Head

After writing the last line of my argument, I realized that simple truth:

"A is a subset of B iff every element in A is in B"

Given your argument, "3" is an element of A yet 3 is neither the set of integers NOR the set containing the set of integers.

Eureka moment. Thanks!
 
  • #4
Going a mile further...

{A} is a subset of B in this example because...

Every element in {A} is in B.

The only element of the set of sets is A. A is the set of all integers. {A} therefore is the subset of B.

------------------------------------------------------------
{{A}} is a subset of B since:

Every element in {{A}} is in B.

The only element in the set of sets of sets is A. The sets of sets of A, {{A}} is an element of B.

That was correct, no?

{{{A}}} is not a subset of B... well because {{A}} is not in B?
 

1. What is discrete math?

Discrete math is a branch of mathematics that deals with discrete objects and structures, as opposed to continuous objects and structures. It includes concepts such as set theory, combinatorics, graph theory, and logic.

2. What are subsets?

Subsets are a fundamental concept in set theory. A subset is a set that contains elements of another set. For example, the set of even numbers is a subset of the set of whole numbers.

3. How are subsets related to discrete math?

Subsets are a key concept in discrete math, as they are used to represent and manipulate discrete objects and structures. For example, in combinatorics, subsets are used to represent combinations of items from a larger set.

4. How are subsets different from proper subsets?

A proper subset is a subset that is not equal to the original set, meaning it does not contain all of the elements of the original set. For example, the set of even numbers is a proper subset of the set of whole numbers, as it does not contain all of the odd numbers in the original set.

5. What are some applications of subsets in real life?

Subsets are used in a variety of real-life applications, such as in computer science for data organization and analysis, in statistics for data sampling, and in finance for portfolio diversification. They are also used in everyday tasks such as organizing a wardrobe or creating a meal plan from a list of available ingredients.

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