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The following paragraph is an excerpt from Discrete Mathematics book of Kenneth Rosen 7edition

The restriction of a universal quantification is the same as the universal quantification of a conditional statement. For instance, ∀x < 0 (x2 > 0) is another way of expressing ∀x(x < 0 → x2> 0). On the other hand, the restriction of an existential quantification is the same as the existential quantification of a conjunction. For instance, ∃z > 0 (z2 = 2) is another way of expressing ∃z(z > 0 ∧ z2 = 2).

Ques : Why universal quantification is same as universal quantification of a conditional statement whereas existential quantification is same as existential quantification of a conjunction?

Please provide proper details. Thank You.

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  • $\begingroup$ I sort of don't know how to answer other than the English interpretation. $\forall x < 0 (x^2 >0)$ would read "for all $x$ less than zero, $x^2$ is greater than zero". In other words, the "statement" is only valid when $x$ is less than zero--for all $x$ less than zero. Thus if we look at all $x$ then the statement is that "if $x$ is less than zero then $x^2$ is greater than zero"...continued... $\endgroup$ – Jared Jul 1 '16 at 7:33
  • $\begingroup$ If we translate e.g. $∃z < 0 (z^2 = 2)$ as $∃z(z < 0 \to z^2 = 2)$ instead of $∃z(z < 0 ∧ z^2 = 2)$, what we get is that $∃z < 0 (z^2 = 2)$ will be true also for the natural numbers. $\endgroup$ – Mauro ALLEGRANZA Jul 1 '16 at 7:34
  • $\begingroup$ On the other hand, $\exists z > 0 (z^2 = 2)$ would translate to "there exists a $z$ greater than zero such that $z^2$ equals 2". In this case we must find (at least one) $z > 0$ such that $z^2 = 2$. Thus to be true we must be able to find a $z$ such that both $z > 0$ and $z^2 = 2$. It would be completely incorrect to write $\exists z (z > 0 \rightarrow z^2 = 2)$. That's easy to satisfy--just choose any $z \leq 0$ then the condition is trivially true. The universal requires a condition for all values and the existential requires the existence of the condition. $\endgroup$ – Jared Jul 1 '16 at 7:37
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Think on these lines.

$(1)$ All humans die.

Equivalent form : For every $x$, if $x$ is human, then $x$ must die. (An implication)

$(2)$. Some animals are color blind.

Equivalent form: There exists some $x$ such that, $x$ is an animal and $x$ is color blind. (A conjunction)

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  • $\begingroup$ That did for me, Thank You. Sometimes answers are just before you and you cannot see them. Thank You $\endgroup$ – Navneet Srivastava Jul 1 '16 at 7:56
  • $\begingroup$ @Navneet Srivastava...my pleasure $\endgroup$ – Nitin Uniyal Jul 2 '16 at 10:51

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