Answer
a) $\forall y \exists x A(x,y)$
b) $\forall x \exists y [B(x,y)\rightarrow C(x)]$
c)$\forall x \exists y D(x,y) \leftrightarrow \exists z E(z)$
d)$\forall y \forall z \exists x \exists a [x\ne a\rightarrow (F(x,y,z) \land F(a,y,z))]$
e) $\forall x [H(x) \rightarrow y G(x,y)] \land \neg \exists x [\neg H(x) \land \forall y G(x,y)]$
Work Step by Step
a) Let A(x.y) mean "consule x is accessible during fault condition y"
$\forall y \exists x A(x,y)$
b) Let B( x. y) mean ”e-mail address of user x has sent message y on the system that is contained in the archive" and C(x) mean "e-mail address of user x can be retrieved".
$\forall x \exists y [B(x,y)\rightarrow C(x)]$
c) Let D(x. y) mean "mechanism x can detect security breach y" and E(z) mean "process z has been compromised".
$\forall x \exists y D(x,y) \leftrightarrow \exists z E(z)$
d)
Let F(x. y. z) mean “path x connects distinct endpoints y and z ”
$\forall y \forall z \exists x \exists a [x\ne a\rightarrow (F(x,y,z) \land F(a,y,z))]$
e) Let G(x. y) means "x knows password of user y" and H(x) mean "x is a system administrator”
$\forall x [H(x) \rightarrow y G(x,y)] \land \neg \exists x [\neg H(x) \land \forall y G(x,y)]$
