Question

Prove the following statement. Assume that all sets are subsets of a universal set U.

Answer 1

Answer:

O U ⊆ A ∪ B                     True

O U ⊆ A ∩ B                    False

O A ∪ B ⊆ U                     True  

O B ⊆ U                            True

O A ⊆ U                            True

O A ∩ B ⊆ U                     True      

Step-by-step explanation:

To prove the first part

$\text{If} \,\,\,\,\,\, A^{c} \subseteq B \,\,\,\,\,\text{then} \,\,\,\,\, A \cup B = U$

Remember that any set is a subset of the universal set. Therefore it is true that

$A \cup B \subseteq U$

Now, given any  $x \in U$   it is true that

$x \in B \,\,\,\, \text{or} \,\,\,\, x \notin B$

Now according to the information given initially

$\text{If} \,\,\,\,\, x \notin B \,\,\,\,\, \text{then} \,\,\,\,\, x \notin A^{c}$

And then you know that    

$\text{If} \,\,\,\,\, x \notin A^{c} \,\,\,\,\,\text{then} \,\,\,\,\,\,\, x \in A$

Therefore    $U \subseteq A \cup B$  and using double inclusion

$U = A \cup B$.

Now using the information just exposed

O U ⊆ A ∪ B                     True

O U ⊆ A ∩ B                    False

O A ∪ B ⊆ U                     True  

O B ⊆ U                            True

O A ⊆ U                            True

O A ∩ B ⊆ U                     True