Prove A is a subset of B with the ELEMENT METHOD

How can we prove that A is a subset of B? The element method has two steps. Firstly, assume that x is an element of A. Then you need to show that x is also an element of B. Because x is chosen generically, this means that every element of A is necessarily an element of B, and thus A is a subset of B. If you want to prove that two sets are EQUAL, then you need to use the element method to show that A is a subset of B AND that B is a subset of A.
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[@Kuda_Christine]

very helpful. thank you

[@mikhailkilianovski8024]

Looks like there is a small typo in example from 2:54 Set Builder notation for B should use m in the condition part, not n

[@Construction-agencyCoUk]

great video i would just say in future please write more clearly - thank you for your effort, much appreciated.

[@RJone89]

I love you Trefor. Thank you.

[@bluejimmy168]

At 6:30, you proved that x is an element of B, which proved that A is subset of B. How do we prove that A is a proper subset of B? The set A has fewer items than B, but i dont know how to write it in a math way. Last question, the method of proof you used in this video is direct proof, right? Thanks.

[@ClaireAlison88]

Yes, it is a direct proof. If you want to say A has fewer items than B, you can say |A|

[@nrgking1667]

Wow it took 6 minutes and I understand this and it took my professor 1 hour and confused a whole bunch of people

[@maxgoals]

nice video , but i could not grasp the idea of putting q=2p
yeah it's logical, but you made a specilization here by making q=2p, what if they p and q equal each other or p is 3q or anything else .. thank you

[@walidibrahimMaxGoals]

@@DrTrefor So p and q are different integers to define different sets so that the first set A have all numbers which are multiple of 4 and the second set B have all numbers which are multiple of 2 .. you need to proof that Set A is a subset B so you need to verify that each element in A is also in B.
There is just one case that makes the numbers in A is subset of B, that's when q=2p .. that means that you must choose an even number for q to make sure that it is in the two sets. If you choose odd number it will be in B but not in A. Hence the set A is subset of B.
You could say also because every number in A is a multiple of 4 that leads to every number of A is also a multiple of 2.
This example is easy but if there is two other complex sets and you want to see if one of them is a subset of the other ?
Thanks

[@royal3537]

Thanks sir

[@ayushregmi4623]

I think P also belongs to positive integer rather than integer only sir , in order to make n a positive integer . or may be I'm wrong😁

[@adrianharo6586]

Nice video, very helpful, do note that you started with nice hand writing but you started accidentally used italic font towards the end.

[@bronsonhill6185]

Why can you let q = 2p?

[@_carrotfx]

because the definition of q is that its an integer, and p was also defined as an integer, 2p would also be an integer because integers are closed under multiplication.

[@michaelmichaels3465]

You're the Mensan version of Mac from IASIP

[@poosywoosy5553]

2p+1 is odd if p is whole. If p is .5, it’s even.

[@jmc042]

do you know what an integer is

[@ninjapirate123]

decimals are not integers

[@iAnarchy89]

i dont understand the letters you use for this and the next few videos, you use n and m to describe the elements of A and B but you use n for both such that statements. is there a reason for that ?
at first i thought you used n for both because A is a subset of B but then i saw you did that for the union and intersection in the next few videos as well.

[@iAnarchy89]

@@DrTrefor so can i say
A = { n ∈ Z | n = 4p, p ∈ Z}
B = { m ∈ Z | m = 2q, q ∈ Z} ?

[@nateburd]

@@iAnarchy89 I think it is incorrect. The set variable should be indicated in the set build.

[@iAnarchy89]

@@nateburd hi I don’t understand, what is the m there for?

[@nateburd]

@@iAnarchy89 the top says n is a in the integers Z and that n will be 4 times some integer p (also an integer).
The bottom says m is an integer such that m will be 2 times q (also an integer).
I am studying maths too but I'm pretty sure there should be an m in the bottom set builder (where in the video is an n). I've emailed the professor and am awaiting a response.

[@iAnarchy89]

@@nateburd ahhh sorry I misunderstood your earlier reply, I thought you meant my comment was wrong haha. Thanks it was bugging me thought my understanding of the concepts were wrong.