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1. The statements given are-
(a)
(b)
(c)
To prove (a) (b) i.e.
Let us assume to the contrary that
Hence,
But, which means that if .
This is a contradiction. So, our assumption is wrong and
To prove (b) (c) i.e.
We know that:
        [Intersection with null set]
            =        [Since, ]
            =        [, by definition]
            =         [Distributive Law]    
            =             [U is Universal set]
            =                 [Intersection with Universal set]
Therefore,
    
To prove (c) (a) i.e.
Let, [Since, ]
        
        
        .
Since, (a) (b) and (b) (c) then it follows from transitivity that (a) (c).
Also, (c) (a) and now (a) (c) then by equivalence (a)(c)
Similarly we can show that (b) (c) and hence (a), (b) and (c) are all logically equivalent.
2. The statements given are-
(a)
(b)
(c)
(c) (b): If then
Consider,
.
(b) (a): If then ,
Similarly,
Thus,
    [Replacing with and with ]
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