Theorem 1: For any three sets , prove that:
i)
(ii) .
Proof:
(i) Let be an arbitrary element of
. Then
[By Definition]
[By Definition of Union]
… … … … … i)
Now, let be an arbitrary element of
.
Then,
… … … … … ii)
Hence from i) and ii) we get
ii) Let be an arbitrary element of
… … … … … i)
Let be an arbitrary element of
… … … … … ii)
Hence, from (i) and (ii), we get .
Theorem 2: For any three sets , prove that:
.
Proof:
Let be an arbitrary element of
. Then
… … … … … i)
Again, let be an arbitrary element of
… … … … … ii)
Hence, from (i) and (ii), we get
Theorem 3: If are any two non-empty sets, then prove that:
.
Proof:
First, let . Then we have to prove that
.
and,
Conversely, let . Then we have to Prove that
.
Let be an arbitrary element of
for all
Again, let be an arbitrary element of
for all
Hence, .
Theorem 4: If , show that
.
Proof:
Let be an arbitrary element of
Then,
Hence,
Theorem 5: If , prove that:
for any set
.
Proof:
Let be an arbitrary element of
.
Then,
Theorem 6: If , prove that:
Proof:
Let be an arbitrary element of
,
Then
Theorem 7: For any sets prove that:
.
Proof:
Let be an arbitrary element of
Similarly,
Hence,
Theorem 8: For any three sets prove that:
(i)
ii)
Proof:
i) [ By De-Morgan’s law ]
ii) [ By De-Morgan’s law ]
Theorem 9: Let be two non-empty sets having
elements in common, then prove that
have
elements in common.
Proof:
We have,
[On replacing
by
by
]
It is given that latex n elements, so
has
elements.
But
has
elements
Hence have
elements in common
Theorem 10: Let be a non-empty set such that
. Show that
.
Proof:
Let be an arbitrary element of
.
Then, for all
for all
Thus,
Let be an arbitrary element of
.
Then, for all
for all
From i) and ii) we get .