Proving or disproving operations on sets

[ver_mathstats]

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.

 

[Mark44]

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.

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]

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?

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]

Yes it is possible for a to not be a an element of B and C as well.

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]

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.

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]

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.

Sure, that works as a counterexample.

 

[Ray Vickson]

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.

You were asked to "prove or disprove", and your example disproves. That's all there is to it!

 

[ver_mathstats]

You were asked to "prove or disprove", and your example disproves. That's all there is to it!

Okay thank you.