Answer
See step by step answer for proof.
Work Step by Step
a) Given : U is the universal set
To proof- $A-\emptyset =A$
Proof-
$A-\emptyset =${$x| x \in A-\emptyset$}
Using the definition of the difference, an element of $A-\emptyset$ is an element that is in $A$ but not in $\emptyset$.
$=${$x|x \in A \land x \notin \emptyset$}
The empty set does not contain any element, thus the statement $ x \notin \emptyset$ is always true.
={$x|x \in A \land T$}
Use the identity law,
={$x|x \in A $}=A
Hence proved.
b) Given : U is the universal set
To proof- $\emptyset -A =\emptyset$
Proof-
$\emptyset-A =${$x| x \in \emptyset-A$}
Using the definition of the difference, an element of $\emptyset-A$ is an element that is in $\emptyset$ but not in $A$.
$=${$x|x \notin A \land x \in \emptyset$}
The empty set does not contain any element, thus the statement $ x \in \emptyset$ is always false.
={$x|F \land x \notin A$}
Use the domination law.
={$x|F$}
The empty set does not contain any element, thus the statement $x \in \emptyset$ is always false.
={$x|x \in \emptyset$}
=$\emptyset$
Hence Proved.
