Proving A Subset B & C $\Rightarrow$ A Subset (B $\cup$ C)

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In summary, the conversation discusses two statements and the task of determining whether they are true or false. The first statement is $A \subset B$ and $A \subset C$ if and only if $A \subset (B\cup C)$, and the second statement is $A \subset B$ or $A \subset C$ if and only if $A \subset (B\cap C)$. The speaker provides a counterexample for each statement, showing that they are in fact false.
  • #1
cbarker1
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Dear Everybody,

I am struggling now for determining if the following statements are true or false. If the statement is true, then prove it. If not, make a counterexample.
Here are the statements:
  1. $A \subset B$ and $A \subset C$ if and only if $A \subset (B\cup C)$.
  2. $A \subset B$ or $A \subset C$ if and only if $A \subset (B\cap C)$.

My attemption:
Let A={1,2,3}, B={1,2,4}, and C={3}.
  1. I believe this is true. $A\subset B$ and $A\subset C$. Thus $A \subset (B\cup C)$ is true.
  2. I don't know how to begin.
Thanks,
Cbarker1
 
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  • #2
Hi Cbarker1.

The two statements are in fact false, thus you just need to find a counterexample to disprove each. Hints:

  1. The $\Rightarrow$ part is true but not the $\Leftarrow$ part. Consider $A\subset B$, $A\ne\emptyset$, and $C=\emptyset$.
    -
  2. This time $\Leftarrow$ is true but not $\Rightarrow$. Consider $A\subset B$, $A\ne\emptyset$, and $B\cap C=\emptyset$.
 

FAQ: Proving A Subset B & C $\Rightarrow$ A Subset (B $\cup$ C)

What does the statement "Proving A Subset B & C $\Rightarrow$ A Subset (B $\cup$ C)" mean?

The statement means that if a set A is a subset of both sets B and C, then A is also a subset of the union of sets B and C.

Why is it important to prove that A is a subset of (B $\cup$ C)?

Proving that A is a subset of (B $\cup$ C) is important because it helps to establish the relationship between the three sets and can provide useful information for solving problems and making decisions.

How do you prove that A is a subset of (B $\cup$ C)?

To prove that A is a subset of (B $\cup$ C), you must show that every element in A is also in (B $\cup$ C). This can be done by using the definition of subset and logical reasoning.

Can A be a subset of (B $\cup$ C) if it is not a subset of both B and C?

No, if A is not a subset of both B and C, then it cannot be a subset of their union. This is because the union of two sets contains all the elements of both sets, so if A is not a subset of either set, it cannot be a subset of their union.

What is the difference between proving A is a subset of (B $\cup$ C) and proving A is equal to (B $\cup$ C)?

Proving A is a subset of (B $\cup$ C) means that all the elements of A are also in (B $\cup$ C), but there may be additional elements in (B $\cup$ C) that are not in A. Proving A is equal to (B $\cup$ C) means that A and (B $\cup$ C) have the exact same elements and there are no additional elements in either set.

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