# Math Help - Find counterexamples to the following statements.

1. ## Find counterexamples to the following statements.

I'm having trouble getting started with the following question:

Find counterexamples to the following statements.

$A \subseteq B \Rightarrow A^c \subseteq B^c$

$(A \not\subseteq B) \wedge (B \not\subseteq C) \Rightarrow (A \not\subseteq C)$

$(A \subseteq B) \wedge (B \not\subseteq C) \Rightarrow (A \not\subseteq C)$

How should I make stuff up?

2. ## Re: Find counterexamples to the following statements.

Originally Posted by maclunian
I'm having trouble getting started with the following question:

Find counterexamples to the following statements.

$A \subseteq B \Rightarrow A^c \subseteq B^c$

$(A \not\subseteq B) \wedge (B \not\subseteq C) \Rightarrow (A \not\subseteq C)$

$(A \subseteq B) \wedge (B \not\subseteq C) \Rightarrow (A \not\subseteq C)$
Let $\mathcal{U}=\{0,1,2,3,4,5,6,7,8,9\}$
If $A=\{2,3,5,6\}$ then $A^c=\{0,1,4,7,8,9\}$.
Now you do some of your own work.

3. ## Re: Find counterexamples to the following statements.

What would be an example to the first one though? How would I notate it?

4. ## Re: Find counterexamples to the following statements.

suppose B = {1,2,3,5,6}. is it true that A⊆B? what can you say about $B^c$?

5. ## Re: Find counterexamples to the following statements.

Originally Posted by Deveno
suppose B = {1,2,3,5,6}. is it true that A⊆B?
Well that would depend on what A = ?

If A = {2,3,5,6} and B = {1,2,3,5,6} then it is true that A⊆B, right?

6. ## Re: Find counterexamples to the following statements.

assume the same A as Plato gave you.

7. ## Re: Find counterexamples to the following statements.

then it is true that A⊆B, right?

8. ## Re: Find counterexamples to the following statements.

So here's my answer for the first one:

$A \subseteq B \Rightarrow A^c \not\subseteq B^c$

because $\{2,3,5,6\} \subseteq \{1,2,3,5,6\} \Rightarrow \{0,1,4,7,8,9\} \not\subseteq \{0,4,7,8,9\}$

Is this correct?

9. ## Re: Find counterexamples to the following statements.

Originally Posted by maclunian
So here's my answer for the first one:

$A \subseteq B \Rightarrow A^c \not\subseteq B^c$

because $\{2,3,5,6\} \subseteq \{1,2,3,5,6\} \Rightarrow \{0,1,4,7,8,9\} \not\subseteq \{0,4,7,8,9\}$

Is this correct?
You tell us if it is correct or not.

10. ## Re: Find counterexamples to the following statements.

Originally Posted by maclunian
So here's my answer for the first one:

$A \subseteq B \Rightarrow A^c \not\subseteq B^c$
whether this statement is correct or not, it is not what you were asked to show. you were asked to find a counter-example, that is, sets A and B for which:

$A\subseteq B \not\implies A^c \subseteq B^c$

so you should have a statement like:

$A \subseteq B$ but $A^c \not \subseteq B^c$,

for some specific A and B (one example, not a general case which the counter-example "proves").