Prove A is not a subset of C, or else B is not a subset of A

[Enh0702]

Let A, B, and C be sets such that A union B is not equal to A intersect C. Prove that A is not a subset of C or B is not a subset of A.

 

[stapel]

Let A, B, and C be sets such that A union B is not equal to A intersect C. Prove that A is not a subset of C or B is not a subset of A.

What have you tried? How far have you gotten?

Since A-union-B is not equal to A-intersect-C, there must be an element "x" in A-union-B that is not in A-intersect-C. Then, by definition of "union", x is either in A or in B.

Suppose x is in A. Since x is not in A-intersect-C, then x cannot be in C. What does this tell you about A being a subset of C?

Suppose x is not in A. Where can you go from there? And so forth.

Eliz.

 

[pka]

This is more complicated than I thought at first.
Given $\displaystyle A \cup B \not= A \cap C$. Assume that $\displaystyle \left[ {A \not\subset C \vee B \not\subset A} \right]$ is false.
Then we have $\displaystyle \left[ {A \subseteq C \wedge B \subseteq A} \right]$ is true.
If \(\displaystyle \left[ {A \subseteq C \right]\) then \(\displaystyle \left[ {A \cap C} \right] = A\).
So from the given $\displaystyle A \cup B \not= A$, that implies that $\displaystyle {B \not\subset A}$ which is in contradiction to the assumption.

Do you see how to finish?