Use Venn-diagram to examine the validity of the following arguments :
(i) `S_(1)` : Natural numbers are integers
`S_(2)` : x is an integer
S: x is a natural number.
(ii) ` S_(1)`: Natural numbers are integers.
`S_(2)` : x is an interger.
S : x is not a natural number.

Explanation

Let N = set of all natural numbers
Z = set of all integers.
(i) The truth of statemnt `S_(1)` is represented by placing the set N entirely inside the set Z i.e. by `N sub Z` , and the truth of statement `S_(2)` is represented by placing a dot labelled, 'x' inside the set Z as shown in Fig .5. This dot may be inside N or it may be outside it also.
Now,
`S_(1) and S_(2)` are true.
` Rightarrow N sud Z ` and x may or may not be in N.
`Rightarrow ` x is not necessarily a natural number.
`Rightarrow` S need not be true.
Hence, the given argument is in invalid.
(ii) Proceeding as above, we find ,we find that x may or may not be a natuaral number. So, the given argument is invalid.