contestada

Prove the following statement. Assume that all sets are subsets of a universal set U.
For all sets A and B, if Ac ⊆ B then A ∪ B = U.
Once you have assumed that A and B are any sets with Ac ⊆ B, which of the following must you show to be true in order to deduce the set equality in the conclusion of the given statement? (Select all that apply.)
O U ⊆ A ∪ B
O U ⊆ A ∩ B
O A ∪ B ⊆ U
O B ⊆ U
O A ⊆ U
O A ∩ B ⊆ U

Respuesta :

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

[tex]\text{If} \,\,\,\,\,\, A^{c} \subseteq B \,\,\,\,\,\text{then} \,\,\,\,\, A \cup B = U[/tex]

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

[tex]A \cup B \subseteq U[/tex]

Now, given any  [tex]x \in U[/tex]   it is true that

[tex]x \in B \,\,\,\, \text{or} \,\,\,\, x \notin B[/tex]

Now according to the information given initially

[tex]\text{If} \,\,\,\,\, x \notin B \,\,\,\,\, \text{then} \,\,\,\,\, x \notin A^{c}[/tex]

And then you know that    

[tex]\text{If} \,\,\,\,\, x \notin A^{c} \,\,\,\,\,\text{then} \,\,\,\,\,\,\, x \in A[/tex]

Therefore    [tex]U \subseteq A \cup B[/tex]  and using double inclusion

[tex]U = A \cup B[/tex].

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