**Question:**

Write the negation of the following statements:

(i) *p*: For every positive real number *x*, the number *x* – 1 is also positive.

(ii) *q*: All cats scratch.

(iii) *r*: For every real number *x*, either *x* > 1 or *x* < 1.

(iv) *s*: There exists a number *x* such that 0 < *x* < 1.

**Solution:**

(i) The negation of statement *p* is as follows.

There exists a positive real number *x*, such that *x* – 1 is not positive.

(ii) The negation of statement *q* is as follows.

There exists a cat that does not scratch.

(iii) The negation of statement *r* is as follows.

There exists a real number *x*, such that neither *x* > 1 nor *x* < 1.

(iv) The negation of statement *s* is as follows.

There does not exist a number *x*, such that 0 < *x* < 1.

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