The discussion centers on the mathematical statement: if A⊆B∪C, then A⊆B or A⊆C. Participants explore the validity of this statement, ultimately concluding that it can be disproven. A counterexample is provided with sets A = {8, 9, 10}, B = {8, 9}, and C = {10, 11}, demonstrating that A is a subset of B∪C but not a subset of either B or C. This confirms that the original statement is false.
PREREQUISITESStudents of mathematics, educators teaching set theory, and anyone interested in understanding logical proofs and counterexamples in mathematics.
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!