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!
The purpose of proving or disproving operations on sets is to ensure the validity and accuracy of mathematical statements involving sets. By proving or disproving operations, we can determine if a given statement or equation involving sets is true or false.
To prove an operation on sets, we must show that the operation holds true for all elements in the sets. This can be done through logical reasoning and mathematical proofs, using properties and definitions of sets and set operations.
Proving an operation on sets involves showing that the operation holds true for all elements in the sets, while disproving an operation involves finding a counterexample that shows the operation does not hold true for all elements.
Some common operations on sets that are often proved or disproved include union, intersection, complement, and subset operations. These operations are fundamental to set theory and are used to manipulate and compare sets.
Proving or disproving operations on sets is important because it allows us to verify the accuracy of mathematical statements involving sets. It also helps us to better understand the properties and relationships between different sets, which can be applied in various fields such as computer science, statistics, and economics.