[Math] True or false. A set is any collection of objects.

elementary-set-theory

  1. True or false. A set is any collection of objects.

  2. True or false. A proper subset of a set is itself a subset of the set, but not vice versa.

  3. True or false. The empty set is a subset of every set.

I need you confirm if my answers are correct or wrong

my effort

1-TRUE. A set is a collection of objects. The objects are referred to as elements of the set.

2-FALSE. A “proper subset” of a set A is defined as a set B that is contained by A, but is not equal to A. If A had a subset B, where B is defined as A, then A=B, and thus does not satisfy the conditions for a proper subset, although it is still always a subset of itself.

3-TRUE. An empty set contains no elements while a subset contains elements that are not in the other comparing set. Hence an empty set becomes a subset of all the other sets because it has no elements and the other set contains elements.

Best Answer

  1. FALSE

A set can't be any collection of objects because certain collections are paradoxical. For example, take that a set is a collection of all sets such that these sets are not members of themselves (Russel's Paradox), and we'll call that set $A$. Then by definition $A$ should include itself in $A$ but if it does so then $A$ contains a member(itself) that is a member of itself. And if $A$ does not include itself then it can't be a set of all set such that these sets do not contain themselves.

Another simpilier paradox is Cantor's paradox in naive set theory that says say that $X$ is the set of all sets, but such a set can't exist because you can always formulate a new set that contains all the elements that $X$ does and the set including $X$ that is ${X}$, so you can't have a set of all sets.

2.TRUE If $A$ is equal to $B$ then all the elements in A are also in B and vice verse. But if $A$ does not contain all the elements $B$ or $B$ does not contain all elements in $A$ only then can one a be a proper subset of the other. So it is true.

  1. TRUE The empty set is a subset of every set. This has to do with the definition of a subset.

Please see the link below if you want to see an explanation

http://mathcentral.uregina.ca/QQ/database/QQ.09.06/narayana1.html

and here if want to see the proof

https://books.google.com/books?id=NuFeW8N2hlkC&pg=PA21&lpg=PA21&dq=proof+the+O+is+subset+of+every+set&source=bl&ots=5TpYc_2MEb&sig=WNghOHNFCJHdw-mlynVH9Dle-uY&hl=en&sa=X&ved=0CFIQ6AEwCGoVChMIsumpqcmrxwIVxRw-Ch2fYw1C#v=onepage&q=proof%20the%20O%20is%20subset%20of%20every%20set&f=false

Related Question