Answer
See step by step answer for proof
Work Step by Step
a) Given : A and B are sets with $A \subseteq B$
To proof: $ A \cup B=B$
Proof-
First Part- Let $ x \in A \cup B$.
Using the definition of the union, we then know that x has to either in set A or in set B.
$$x \in A \lor x \in B$$
Since $A \subseteq B$ , $ x\in A$ $\implies$ $x \in B$
$$x \in B \lor x \in B$$
Using the idempotent law,
$$x \in B$$
We have shown that $ A \cup B \subseteq B$.
Second part- Let $x \in B.$
Use addition law,
$$x \in B \lor x \in A$$
Using commutative law:
$$x \in A \lor x \in B$$
Use the definition of the union:
$$x \in (A \cup B)$$
We have shown that $ B \subseteq (A \cup B)$
Conclusion: Since $ A \cup B \subseteq B$ and $B \subseteq A \cup B$,
Thus, $ A \cup B = B$
b) To proof - $ A \cap B =A$
Proof-
First Part- Let $ x \in A \cap B$
Using the definition of the intersection, we know that x has to be in both sets, $$x \in A \land x \in B$$
Using simplification,
$$ x \in A$$.
By the definition of a subset, we have then shown that $ A \cap B \subseteq A$
Second part- Let $x \in A$
Since A is a subset of B, $$x \in B$$
Use the conjuction, $$ x \in A \land x \in B$$
Using the definition of the intersection, we know that x is in the intersection when x is in both sets:
$$x \in A \cap B$$.
We have shown that $A \subseteq A \cap B$.
Since $A \cap B \subseteq A$ and $ A \subseteq A \cap B$,
$A \cap B = A$.
