Proof: A is a Subset of B iff A intersect B Equals A | Set Theory, Subsets

A is a subset of B if and only if A intersect B equal A. We will prove this set theory result in today's video set theory lesson!
The proof is straightforward and follows easily from definitions. Always good to get some practice learning how to use our fundamental set theory definitions to prove subset relations and to prove that sets are equal!
Proof of the analogous result for set union: • Proof: A is a Subset o...
I hope you find this video helpful, and be sure to ask any questions down in the comments!
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Vallow Bandcamp: vallow.bandcamp.com/
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Vallow SoundCloud: / benwatts-3
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[@nazlkalkan9457]

I appreciate that you are doing this :)

[@WrathofMath]

It is my pleasure! Thanks for watching and let me know if you ever have any questions!

[@brunourbancedron9795]

Im studing phisics in Spain and this video was very helpful, u are the best

[@WrathofMath]

So glad to help, thanks a lot for watching!

[@kimhughes1147]

Kudos mate - excellent work

[@WrathofMath]

Thanks a lot!

[@dunkelheit3129]

Thank you so much for the work you do! I have a little doubt about logic at 3:45. I know that a set A being a subset of a set B means "a in A implies a in B", so at that point in the proof when we take a in A we have the information "a in A implies a in B", so we still don't have "a in A and a in B". By the logic point of view, being an element of A intersection B means "a in A and a in B", so I was wondering how we can deduce that the "and" requested in the definition of intersection is true from that implication to conclude the proof. My reasoning is the following: since we know that "a in A" is true by hypothesis and we know (from the table of truth) that an implication with the first proposition true must have the second proposition also true to be a true implication then "a in B" must be true as well because the implication is true by hypothesis (it follows from assuming A subset of B). So we have both the proposition "a in A" and "a in B" true, which is the logic definition of intersection of A and B because "a in A and a in B" is true only when both "a in A" and "a in B" are true. Is this correct? Thanks!

[@WrathofMath]

You're very welcome, it is my pleasure to create these lessons and thank you for watching! Your reasoning is correct, I was saying the same thing just with less of the step by step details. We assume A is a subset of B. We take an arbitrary element of A. By definition of subset this element is also in B. Thus, the element is in A and it is in B, which by definition of set intersection means it is in A intersect B. Hope that helps!

[@rahmatazam3281]

@@WrathofMath "Thus, the element is in A and it is in B, which by definition of set intersection means it is in A intersect B."
That implication could also mean A union B, no? Why do we take intersection specifically

[@user-uu9xr1wt9i]

Thank you so much, now I got it... Keep up the good work 😊

[@WrathofMath]

Glad it helped - thanks for watching!

[@meirabasinam4205]

Thank u wrath of math... This video helps me a lot... Please keep continuing to upload Such kind of proof and solutions... and problems too mainly from the Real analysis (Robert G. Bartle)

[@WrathofMath]

You're very welcome Meiraba, thanks for watching and I am glad it helped! Let me know if there are any problems specifically you'd like to see from that text; I hope to do more on real analysis as time permits!

[@azizhani33]

A is a subset of B means if a belongs to A then a belongs to B so a belongs to both A and B and therefore belongs to their intersection.
A intersect B equals A means if a belongs to A then a also belongs to the intersection and therefore to B as well. So to recap, if a belongs to A then a also belongs to B, which means by definition that A is a subset of B.

[@avikasingh8558]

Thanks a lot sir, by using venn diagram u made it easy to understand

[@WrathofMath]

Glad it helped! You're welcome and thanks for watching!

[@deandustin3503]

I wish I was in a proofs class so that I could get all my homework answers from you.

[@WrathofMath]

I wish I was too

[@zazas8573]

Be better

[@pramishapaudel9965]

Thank u so much for this video!!!!!!

[@WrathofMath]

My pleasure, thanks for watching!

[@hamzehtbakhe3585]

thats help me a lot ...thank u very much♥️♥️

[@WrathofMath]

My pleasure!

[@sivapriya.s6549]

Clear sir, good job

[@WrathofMath]

Thank you, I am glad it was clear!

[@supriyadebnath1159]

thank you sir

[@WrathofMath]

Glad to help!

[@kushagrasharma6679]

At 4:10, how did A become from only one element of A?
X is only an element of A, so how ?

[@WrathofMath]

Thanks for watching and for the question! I'm not entirely sure what you mean. Are you wondering how we can say that A is a subset of A intersect B, when we have only demonstrated that ONE element of A is in A intersect B? If so, the reason is that x is an arbitrary element of A. Thus, given any element of A, the same arguments apply. So we have sufficiently justified that every element of A is an element of A intersect B. Because the same argument works for every element of A. Does that answer your question?

[@kushagrasharma6679]

@@WrathofMath Yes, that was my question. Thank you for answering:)

[@angellj6412]

Need help Sir. What if A intersect B equal to B? Will B a subset of A?

[@AmitKumar-zg3zh]

I wish that I also have teacher like you

[@WrathofMath]

At least I can help via the internet! Thanks for watching, and let me know if you ever have any questions!

[@aanya6275]

If A C(subset of) B then prove that A x A = (A x B) intersection (B x A)

[@user-uu9xr1wt9i]

I don't understand yet

[@Agustinoism]

patrick jmt leave you tube a new king has arrived

[@rizwanmeher1226]

A subset of B if and only if A intersection complement of B is equal to phi plz prove this

[@WrathofMath]

Thanks for watching and here is a proof of that result, stated slightly differently: th-cam.com/video/y14VGV9G6iA/w-d-xo.html

[@smallyoutuber2520]

Love from INDIA

[@WrathofMath]

Thank you for your support! Much love back from the east coast of the USA!

[@smallyoutuber2520]

@@WrathofMath I'm from INDIA
And my language is Assamese