# Logic - finding most general statement

Let A and B be sets. For each statement below, please write down the most general statements you can make about A and B. Make sure you justify your answer.

a. A ∪ B = A?

b. A ∩ B = A?

c. A ∪ B = A ∩ B?

d. A − B = A?

e. A − B = B − A?

a.B= $\varnothing$ or B$\subseteq$ A

b. B= U

c. A=B

d. A ∩ B = $\varnothing$

e. A=B

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• a. $A \cup B = A.\quad$

Here, your conclusion that $B\subseteq A$ suffices.

Since the empty set is a subset of every set, so even if it is the case that $B = \varnothing$, that is covered by "$B\subseteq A$".

• b. $A \cap B = A$

From (b) you can conclude $\;A\subseteq B$.

(It's not necessarily the case that $B = U$, where $U$ is the universal set; you can, however, write $A \subseteq B \subseteq U$.)

• c. $\to$ e. These look good to me!

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Thanks Man,,,,, –  Hooman Dec 13 '12 at 18:25
Your welcome, Hoorman. Do you see why (b) means $A\subseteq B$? –  amWhy Dec 13 '12 at 18:29
Sure , U = Universe ,Thanks for the help –  Hooman Dec 13 '12 at 18:31