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!

I hope you find this video helpful, and be sure to ask any questions down in the comments!

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The outro music is by a favorite musician of mine named Vallow, who, upon my request, kindly gave me permission to use his music in my outros. I usually put my own music in the outros, but I love Vallow's music, and wanted to share it with those of you watching. Please check out all of his wonderful work.

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+WRATH OF MATH+

Follow Wrath of Math on...

[deandustin]

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

[dunkelheit]

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!

[aanya]

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

[azizhani]

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.

[avikasingh]

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

[brunourbancedron]

Im studing phisics in Spain and this video was very helpful, u are the best <3

[meirabasinam]

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)

[Agustinoism]

patrick jmt leave you tube a new king has arrived

[nazlkalkan]

I appreciate that you are doing this :)

[kushagrasharma]

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

[rizwanmeher]

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

[rauhashoombe]

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

[hamzehtbakhe]

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

[AmitKumar-zgzh]

I wish that I also have teacher like you