Prove that A is a subset of A union B for any sets A and B.

Here we are proving another theorem within set theory. This theorem is covered in "A Transition to Advanced Mathematics," by Smith, Eggen, and St. Andre.

[@saravana4217]

Please Clear me one thing... Clearly,x is an element of A then Why you are using x might be an element on B(I mean x also belongs to B) ...Why?

[@snellbrosmath]

The line "x in A implies x in A or x in B?"

[@saravana4217]

@@snellbrosmath yeah

[@snellbrosmath]

Well, T=T or F. Also, T=T or T. So regardless of if "x in B" is true or false, if "x in A" is true, then "x in A or x in B" is true.

[@firstname4337]

why can't you just write one line and say "that is the definition of what U is"

[@snellbrosmath]

Can you give me a sample argument?

[@TheSeriousSean]

Because that’s hand waving and this appears to be introductory set theory where many students learn rigorous proofs for the first time.

[@missemotional8710]

Couldn't we do it using Venns diagram? Where the A and B cricle intersect it denominates the union of two groups. In that case , some of the A's values are withing the unions intersection which would automatically make A a subgroup of A U O, correct me if I'm wrong or if theres a flaw in my logic.

[@snellbrosmath]

@missemotional8710 Venn Diagrama and pictures are great to help aid in understanding and visualizing a proof. However, pictures can not be the entire proof.

[@roflattheworld]

Brain is a bit wine-addled atm, but off-the-cuff I like the following: if x not in A and x not in B then x not in A since this above contrapositive is trivially true, then the statement is true

[@jordancaasi5361]

i think the original proposition is just as trivial lol

[@roflattheworld]

@jordancaasi5361 for sure - am just a big fan of the contrapositive haha