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Express each of these system specifications using predi-
cates, quantifiers, and logical connectives, if necessary.

e) No one knows the password of every user on the sys-
tem except for the system administrator, who knows

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In the universe of all system users and administrators.

• P(x,y) - x knows y's password
• U(x) - x is an user of system
• S(x) - x is a system administrator

Given statement is equivalent to -

1. $\forall$x ( $\forall$y ($\neg$ P(x,y)) $\lor$ S(x) ) $\implies$ No one knows the password of every user on the system except for ths system administrator.
2. $\forall$x ( S(x) $\implies$ $\forall$y (P(x,y)) ) $\implies$ System administrator knows all passwords.

Thus, given statement is -
$\forall$x ( $\forall$y ($\neg$ P(x,y)) $\lor$ S(x) ) $\land$ $\forall$x ( S(x) $\implies$ $\forall$y (P(x,y)) )