Prove that if |A| ≤ |B| and |B| ≤ |C| then |A| ≤ |C| (1 Viewer)

[sarah666]

Hello,
Can someone help me on this question please ?​

​

Prove that if |A| ≤ |B| and |B| ≤ |C| then |A| ≤ |C|

Thanks in advance.​

 

[Trebla]

|A|≤ |B|
|B| ≤ |C|
So, |A| ≤ |B| ≤ |C|
hence |A| ≤ |C|

 

[conics2008]

|A| ≤ |B| and |B| ≤ |C| then |A| ≤ |C|

You said if A is less than or equal to B and if B is less than or equal to C then

A must be less than or equal to C because if you say B = C

then A less than or equal to C ..

What kind of maths questoin is this... ?

 

[Affinity]

they could mean cardinalities..

 

[lolokay]

(leaving out |'s)

A ≤ B, B≤ C
let B = A + x
let C = B + y
C = A + x + y
A≤ A+x
0≤ x
B≤ B + y
0≤ y
0≤ x≤ x + y
0≤ x + y *
A≤ A + x + y
A≤ C

*although, given this step you may as well just use A≤ B≤ C, so A≤ C

edit: deleted below post since it doesn't apply, assuming cardinality is meant

 

[vds700]

|A|≤ |B|
|B| ≤ |C|
So, |A| ≤ |B| ≤ |C|
hence |A| ≤ |C|

this seems the most simple and logical proof

 

[Affinity]

|A| ≤ |B| could mean "there exist an injection from set A into set B"

 

[sarah666]

|A| ≤ |B| could mean "there exist an injection from set A into set B"

If this is the case, would the proof be any different from the proofs given by the posts above?

Also, thanks everyone for the help so far.

 

[Affinity]

would be slightly different.. by the way, where did you get that question from

 

[Slidey]

Look up the property of transitivity, sarah.

 

[sarah666]

would be slightly different.. by the way, where did you get that question from

Oh so it is different, can you please show me the proof ? Because I have absolutely no idea.

This question was given to me by a friend of mine and she doesn't know how it is done as well. :-(. If you can help me, I wolud be very appreciated.

 

[darkliight]

Let f:A->B and g:B->C be injective (these exist by assumption). Then gof (g after f) is injective. To see this, suppose gof(x) = gof(y), then f(x) = f(y) since g is injective, and x = y since f is injective.

This surely isn't what your teacher wanted though, ie, your teacher probably wanted something like the first couple of posts here.

 

[kooltrainer]

wahh?.. wtf is injection from B into A.. wah??
dw, not in syllabus xD