1. Probability Notation

What is the difference between ⊆ and ⊂ ?

I am currently using Probability and Statistics: Crash Course. It told what ∪ and ∩ meant, but not ⊂. Strange. I gather that the latter means "subset of", but I don't know what ⊆ means at all.

It says that Theorem 1-1 is:
"If A1 ⊂ A2, then
P(A1) <= P(A2) and P(A2-A1) = P(A1) - P(A2)"

The above is as written, double-checked (except that the numbers are in subscript, and <= is a single character).

The first part sounds right, but what about the second part? If A1 is a subset of A2, then shouldn't P(A1) - P(A2) be a negative number?

And what is with subtracting A1 from A2 in parenthesis? I didn't think events necessarily had mathematical values, though probabilities of events obviously do.

2. Originally Posted by Boris B
What is the difference between ⊆ and ⊂ ?

I am currently using Probability and Statistics: Crash Course. It told what ∪ and ∩ meant, but not ⊂. Strange. I gather that the latter means "subset of", but I don't know what ⊆ means at all.

It says that Theorem 1-1 is:
"If A1 ⊂ A2, then
P(A1) <= P(A2) and P(A2-A1) = P(A1) - P(A2)"

The above is as written, double-checked (except that the numbers are in subscript, and <= is a single character).

The first part sounds right, but what about the second part? If A1 is a subset of A2, then shouldn't P(A1) - P(A2) be a negative number?

And what is with subtracting A1 from A2 in parenthesis? I didn't think events necessarily had mathematical values, though probabilities of events obviously do.
Maybe a simple example will help on a six sided die
The sample space is

$S=\left[ 1,2,3,4,5,6\right]$

Let $A_1$ be rolling an odd

$A_1=\left[ 1,3,5\right]$

let $A_2$ be rolling a one

$A_2=\left[ 1\right]$

so $A_2 \subset A_1$

$A_1-A_2=\left[ 3,5 \right]$

$P(A_1-A_2)=\frac{2}{6}$

but

$P(A_1)-P(A_2)=\frac{3}{6}-\frac{1}{6}=\frac{2}{6}$

3. What you're saying makes sense....

I think there is a typo in the book!

Does the ⊆ notation mean "the two sets may be equivalent, or the first is a subset?"

4. Originally Posted by Boris B
What is the difference between ⊆ and ⊂ ?

I am currently using Probability and Statistics: Crash Course. It told what ∪ and ∩ meant, but not ⊂. Strange. I gather that the latter means "subset of", but I don't know what ⊆ means at all.

It says that Theorem 1-1 is:
"If A1 ⊂ A2, then
P(A1) <= P(A2) and P(A2-A1) = P(A1) - P(A2)"

The above is as written, double-checked (except that the numbers are in subscript, and <= is a single character).

The first part sounds right, but what about the second part? If A1 is a subset of A2, then shouldn't P(A1) - P(A2) be a negative number?

And what is with subtracting A1 from A2 in parenthesis? I didn't think events necessarily had mathematical values, though probabilities of events obviously do.
Hi Boris,

I think there is a misprint in your book and the theorem should read

"If A1 ⊂ A2, then
P(A1) <= P(A2) and P(A2-A1) = P(A2) - P(A1)"

⊂, as in A⊂B, means "is a subset of". I think some books use this symbol to mean A is a proper subset of B (meaning A is not equal to B), and they use the symbol ⊆, as in A ⊆ B, to mean that A is a subset of, or equal to, B, but I think this usage is not common. Most of my textbooks just use ⊂ to denote "is a subset of" so A ⊂ B includes the possibility that A=B.

The minus in A-B, where A and B are sets, is the "set difference" operator. A-B is the set of all elements which are in A but not in B. Some texts use a different symbol, \, for this operation, as in A \ B.