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p:  is true for every real numbers and . q: There exists real numbers and  for which . The negation of p is: There exists no real numbers x and y for which which is not equal to q. Hence the given pair of statements are not negation of each other.
Given, p: There exists a capital for every state in India. Quantifier is "There exists". Negation is, p': There does not exist a capital for every state in India. Or, There exists a state in India which does not have a capital.
Given, p: For every real number , is less than . Quantifier is "For Every". Negation is, p': There exists a real number x such that x is not less than x + 1.
Given, p: There exists a number which is equal to its square. Quantifier is "There exists". Negation is, p': There does not exist a number which is equal to its square.
The connecting word here is 'and'. The component statements are: p: x = 2 is a root of the equation . q: x = 3 is a root of the equation .
The connecting word here is 'Or'. The component statements are: p: Square of an integer is positive. q: Square of an integer is negative.
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