The discussion revolves around the statement: if A⊆B∪C, then A⊆B or A⊆C. Participants are exploring the concepts of set theory, specifically subsets and unions.
There is an ongoing exploration of the statement with participants questioning the assumptions involved. A counterexample has been proposed, which some participants acknowledge as a valid approach to disproving the statement.
Participants are working within the constraints of a homework assignment that requires proving or disproving the given statement. There is a focus on understanding the definitions and relationships between the sets involved.
Start by assuming that ##a \in B \cup C##. Is it possible for a to not be an element of B, and also not be an element of C?ver_mathstats said:Homework Statement
Prove or disprove: if A⊆B∪C, then A⊆B or A⊆C.
Homework Equations
The Attempt at a Solution
I am unsure of how to go about proving this. I know that A is a subset of B union C then A is a subset of B or A is a subset of C and I understand what a subset is and what a union is. But I just do not understand how to go about doing this proof.
Thank you.
Yes it is possible for a to not be a an element of B and C as well. Could I disprove this with a counter example?Mark44 said:Start by assuming that ##a \in B \cup C##. Is it possible for a to not be an element of B, and also not be an element of C?
How so? Can you draw a Venn diagram that illustrates this? IOW, ##a \in B \cup C##, but a is not in B and a is not in C.ver_mathstats said:Yes it is possible for a to not be a an element of B and C as well.
Sorry I think I am misunderstanding. I thought of A = {8, 9, 10}, B = {8, 9}, and C = {10, 11}. A would be a subset of B union C, but it wouldn't be a subset of B or of C so I had thought that perhaps I could use that but I don't know.Mark44 said:How so? Can you draw a Venn diagram that illustrates this? IOW, ##a \in B \cup C##, but a is not in B and a is not in C.
Sure, that works as a counterexample.ver_mathstats said:Sorry I think I am misunderstanding. I thought of A = {8, 9, 10}, B = {8, 9}, and C = {10, 11}. A would be a subset of B union C, but it wouldn't be a subset of B or of C so I had thought that perhaps I could use that but I don't know.
ver_mathstats said:Sorry I think I am misunderstanding. I thought of A = {8, 9, 10}, B = {8, 9}, and C = {10, 11}. A would be a subset of B union C, but it wouldn't be a subset of B or of C so I had thought that perhaps I could use that but I don't know.
Okay thank you.Ray Vickson said:You were asked to "prove or disprove", and your example disproves. That's all there is to it!