Chapter  3, Proofs
Section  3.7  More Examples of Proofs
Summary
It contains few more examples to outline proofs which may contain multiple strategies.
Soln1
Existence:
Suppose .
Proof for: :
Suppose . Suppose , then . Since is arbitrary, we can say . Since , it follows that . Thus we have, if , then . Since is arbitrary, it follows that .
Proof for: :
Suppose is arbitrary. Suppose . Suppose , or . Then there must exist some set, say such that . Since , it follows that . Thus . Thus we have . Since is arbitrary, we can conclude .
Uniqueness:
Suppose is a set such that and are true. Now since this is true for all possible values of and we know that if then is true. Thus if , it follows that , or . Now suppose . Then there exists some set, say such that . But since , it follows that , or . Thus . Since is arbitrary, it follows that . Thus we have . It follows that .
Soln2
Existence:
Suppose . Thus we have:
We know that every power set contains a set with empty element, or . Thus one possible value of and it clearly satisfies the above statement.
Uniqueness:
Suppose if , then,
is equivalent to:
This can be true only in following cases:

:
Clearly this is not possible. 
: Clearly this is also not possible.
Thus it contradicts the assumption that such set exists.
Thus there is only one such set .
Soln3
We shall prove this as follows :
 if (a) then (b).
 if (b) then (c).
 if (c) then (a).
where (a), (b) and (c) are the parts given in the problem.
 if (a) then (b).
Suppose (a), is true. Thus is true.
We will prove this part : by contradiction. Suppose . Suppose . Since , it follows that . Similarly, since , it follows . Thus . But this contradicts with the assumption . Thus we have if , then . Since is arbitrary, we can conclude .
Now we will prove part : by contradiction. Suppose . Suppose . It follows that , which is equivalent to . Since , it follows that . Similarly, since , it follows that . Thus , which contradicts our assumption. Thus if , then . Since is arbitrary, we can say that .
 if (b) then (c).
Suppose (b) is true. Thus is true. Suppose , then we have following possible cases:

Case 1: , , : This case is not possible because we know that if , then .

Case 2: , , : Thus . It follows that .

Case 3: , , : Thus . It follows that .

Case 4: , , : This case is not possible because we know that if , then .
Thus from all the cases, if , then . Since is arbitrary, we can conclude that .
 if (c) then (a).
Suppose (c) is true. Thus is true. Suppose . It follows that . Thus we have following cases:

Case 1: , : In this case . It follows that .

Case 2: , : This case is not possible as is not possible.

Case 3: , : This case is also not possible as is not possible.

Case 4: , : Since , it follows that .
Thus from all the cases we have, . Since is arbitrary, we can conclude that .
Soln4
Suppose . Thus if , then . Since , it follows that . Thus exists in atleast one of the sets, say in . Thus we have . Now suppose , then . And since , we can conclude that . Since is arbitrary, we can conclude that .
Soln5
(a)
Suppose . Thus we have:
Since ,
Since is arbitrary, we can conclude .
(b)
Suppose . Thus we have:
Since ,
Since is arbitrary, we can conclude .
(c)
Suppose . Thus exist in atleast one of the sets, say , such that . Thus exist in all the sets of . Thus for any set, say , we have . Thus if , then because . Since can be any set such that , it follows that exists in all the sets such that . Thus also exists in . Since is arbitrary, we can conclude that .
Counterexample for: :
Suppose
Suppose , , .
Thus
And .
Thus .
(d)
We shall prove the theorem .
Suppose . Thus must exist in atleast one of the set, , such that . Since , and , it follows that must exist in . Thus we can conclude that if , then .
Suppose
Suppose , , .
Thus
And .
Thus .
Soln6
Suppose is arbitrary. Suppose . Suppose is an arbitrary real number and .
Then we have: , is equivalent to:
Since , thus
Since :
Thus,
Soln7
Suppose that . Thus for all , there exists a such that for all , if , then
.
Since , if , then:
.
.
.
Thus .
Soln8
Suppose that . Thus for all , there exists a such that for all , if , then
.
.
.
Since , .
Thus, if , then , where .
Thus .
Soln9
The proof and theorem both are correct. Following strategies are used:
 To prove a goal of the form of . Here existential instantiation is used.
 To prove a goal of the form of , all possible cases are considered.