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.
Best Answer
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)