My question is this: If two sets are equivalents does that mean that they are equal. I say no but I'm not sure how to explain my answer can you help me? Thanks.
I quote Bob's answer in
http://www.jiskha.com/display.cgi?id=1244122467
" equivalent sets contain the same number of elements. sets a,b and b,c are equivalent.
Equal sets contain the same number and same elements a,b and b,a are equal
Set A and D are equivalent, but not equal
Set A and C are equal"
Now,
P={1,2,3,4}
Q={A,B,C,D}
Is P equal to Q?
Is P equivalent to Q?
Yes P is equal to Q but P is not equivalent to Q.
> equivalent sets contain the same number of elements.
Do P and Q contain the same number of elements?
> Equal sets contain the same number and same elements
Do P and Q have the same elements?
Yes Pis equivalent to Q and P is equal to Q.
> Equal sets contain the same number and same elements
"same elements" means each member is identical.
{1,2} and {3,4} have the same number of elements, so they are equivalent.
Since 1 does not equal 3, nor 4, so the elements in {1,2} are NOT the same as those in {3,4}, so they are not equal.
Do P and Q have the same elements?
Now what can you say about
A. {1,2,3}, {1,2,3,4}
B. {1,2,3}, {1,2,A}
C. {2,4,6}, {4,6,2}
No P and Q does not have the same elements. That A is equilavent but not equal. B is not equilavent or equal. C is equivalent as well as equal.
A. {1,2,3}, {1,2,3,4}
Do they have the same NUMBER of elements?
No. So the two sets are not equivalent.
If they are not equivalent, they cannot be equal.
B. {1,2,3}, {1,2,A}
Do they have the same NUMBER of elements?
Yes, they both have 3 elements. So they are equivalent.
Are they equal?
Examine the elements one by one. If there is one that is not found in the other, they are not equal.
1=1
2=2 but
3 is not the same as A,
so the two sets are equivalent (have 3 members each) but not equal (at least one member is different)
C. {2,4,6}, {4,6,2}
Do they have the same number of elements?
Yes, so they are equivalent.
Do they have the same elements?
2=2
4=4
6=6
Yes, they have the same elements (order is not important here), so the two sets are equal.
Finally, you can note the following observations:
Two sets can be equivalent but not equal.
If two sets are not equivalent (don't have the same NUMEBR of members), they cannot be equal.