2

I'm learning about Sets. I'm confused in two statements given in my book,i.e
every set is a subset of itself and the empty set is the subset of every set. These two subsets are called improper subset.

Another statement: a subset A of a set B is called proper set of B if A is not equal to B.
I didn't get how phi is improper subset since it is not equal to any non-empty set. I searched about this question in this site but I didn't understand it fully. Please explain. Thankyou in advance.

Mauro ALLEGRANZA
  • 94,169
Yogesh Tripathi
  • 629
  • It's only an "informal" expression and not a formal one. Intuitively, we think at a subset of a set $B$ as a set $A$ built up picking some of the elements of $B$. Thus $B$ itself and $\emptyset$ are "degenerate" cases , obtained by choosing respectively all and none of the elements of $B$. – Mauro ALLEGRANZA Oct 06 '15 at 11:42
  • Note that phi ($\phi$) is not the symbol for the empty set, which is $\emptyset$ or $\varnothing$. – mrp Oct 06 '15 at 11:44
  • @mrp Sometimes its the author being lazy and using $\phi$ instead of $\varnothing$. – Ali Caglayan Oct 06 '15 at 15:40
  • @Yogesh please give reference of your book. thanks. – Hamza Rashid May 15 '18 at 19:56

1 Answers1

3

Authors differ a bit on their definitions here. A "proper subset" of $A$ is always(?) not $A$ itself, but $\emptyset$ may (albeit rarely) be referred to as "proper". The idea is that when we select a subset, we often want to deal with some elements of it; if we selected the empty subset, we can't deal with its elements, so we don't really want to call it "proper".

The word itself doesn't really matter - as long as you know whether a theorem intends to exclude the empty set or include it, it doesn't matter whether you call $\emptyset$ a proper subset of anything or not. When writing your own theorems and proofs, it is good practice to state "a nonempty proper subset" when you mean "a subset which is neither $A$ nor $\emptyset$", or "a strict subset, possibly empty" when you mean "a subset which is not $A$". As long as you define your terms clearly before you use them, you can use whatever terms you want.

Patrick Stevens
  • 36,135