Question

What can you say about the sets A and B if we know that: a) A ∪ B = A? b) A ∩ B = A? c) A − B = A? d) A ∩ B = B ∩ A? e) A − B = B − A?

Answer 1

(a) Given:$A \cup B=A$

Using the definition of the union, an element of $A \cup B$ is an element that is in A or in B.

$A \cup B=\{x \mid x \in A \vee x \in B\}$

Let x be an element of B.

$x \in B$

By the definition of the union, $x$ then has to be an element of $A \cup B$ .

$\Rightarrow x \in A \cup B$

By $A \cup B=A$ , we then know that x also has to be an element of A :

$\Rightarrow x \in A$

Every element of B is thus also an element of A. By the definition of a subset, we then know that B is a subset of A

$B \subseteq A$

(b) Given: $A \cap B=A$

Using the definition of the intersection, an element of $A \cup B$ is an element that is in A and in B.

$A \cap B=\{x \mid x \in A \wedge x \in B\}$

Let $x$ be an element of A.

$x \in A$

By $A \cap B=A$ , $x$ then has to be an element of $A \cap B$ .

$\Rightarrow x \in A \cap B$

By the definition of the intersection $x$ then has to be an element of B as well.

$\Rightarrow x \in B$

Every element of A is thus also an element of B. By the definition of a subset, we then know that A is a subset of B.

$A \subseteq B$

(c) Given: $A-B=A$

Using the definition of the difference, an element of $A-B$ is an element that is in A and not in B.

$A-B=\{x \mid x \in A \wedge x \notin B\}$

Let $x$ be an element of B.

$x \in B$

By the definition of the difference, $x$ then is not an element of $A-B$ ,

$\Rightarrow x \notin A-B$

By $A-B=A$ , we then know that $x$ also cannot be an element of A :

$\Rightarrow x \notin A$

Every element of B is thus also an element of the complement of A. By the definition of a subset, we then know that B is a subset of $\bar{A}$ .

$B \subseteq \bar{A}$

Since the intersection of A and $\bar{A}$ is empty and $B \subseteq \bar{A}$ , the intersection of A and B then also has to be empty.

$A \cap B=\emptyset$

(d) Commutative law for set identities:

$A \cap B=B \cap A$

The given statement is the commutative law and is thus true for all sets A and B. This then means that we cannot say anything about the sets (since there are no conditions in using the commutative law).

(e) Given: $A-B=B-A$

Using the definition of the difference, an element of $A-B$ is an element that is in A and not in B.

$A-B=\{x \mid x \in A \wedge x \notin B\}$

Using the definition of the difference, an element of $B-A$ is an element that is in B and not in A.

$B-A=\{x \mid x \in B \wedge x \notin A\}$

Let us assume $x \in A-B$ . Then $x \in A$ is true and $x \notin A$ is false. By the definition of $B-A,$ $x$ can then not be in $B-A$ . Since none of the elements of $A-B$ are in $B-A$ and since $A-B=B-A$ , the differences then have to be the empty set.

$A-B=B-A=\emptyset$

The difference A-B does not contain any elements, if all elements of A are also an element of B.

$A \subseteq B$

The difference B-A does not contain any elements, if all elements of B are also an element of A.

$B \subseteq A$

Since $A \subseteq B$ and $B \subseteq A$ , the two sets then have to be equal.

$A=B$