Answer to Question #308446 in Discrete Mathematics for Blue

Question

Translate each of these statements into logical expressions using predicates, quantifiers, and logical connectives.

a) No one is perfect.

b) Not everyone is perfect.

c) All your friends are perfect.

d) At least one of your friends is perfect.

e) Everyone is your friend and is perfect.

f) Not everybody is your friend or someone is not perfect. Translate each of these statements into logical expressions using predicates, quantifiers, and logical connectives.

a) No one is perfect.

b) Not everyone is perfect.

c) All your friends are perfect.

d) At least one of your friends is perfect.

e) Everyone is your friend and is perfect.

f) Not everybody is your friend or someone is not perfect.

Answer 1

Solution

Let the domain consist of all people

Consider $p(x)$ as proposition function, where "$x$ ", is perfect.

Consider $q(x)$ as proposition function, where "$x$ ", is your friend or in your class.

Solution (a) No one is perfect.

$∀x(¬p(x))$

Solution (b) Not everyone is perfect.

$¬∀x(p(x))$

Solution (c) All your friends are perfect.

$∀x(q(x)\rightarrow p(x))$

Solution (d) At least one of your friends is perfect.

$\exists x(q(x)\rightarrow p(x))$

Solution (e) Everyone is your friend and is perfect.

$∀x(p(x))∧∀x(q(x)))$

Solution (f) Not everybody is your friend or someone is not perfect.

$¬∀x(q(x))\vee (\exist x¬(p(x))))$