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?
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.
7 answers
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?
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}
"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.
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.