Question

In this question, the domain of discourse is a set of employees who work at a certain company. Ingrid is one of the employees at the company. Define the following predicates: S(x): x was sick yesterday W(x): x went to work yesterday V(x): x was on vacation yesterday Translate the following English statements into a logical expression with the same meaning.

(a) Everyone was well and went to work yesterday.

(b) Everyone who was sick yesterday did not go to work.

(c) Yesterday someone was sick and went to work.

(d) Someone who missed work was neither sick nor on vacation.

(e) Ingrid was sick yesterday but she went to work anyway.

Answer 1

Logical predicates are used to describe properties or actions of employees at a company, translating English statements into formal logical expressions. We defined the predicates S(x), W(x), and V(x), and used them to translate several statements about the employees' conditions on a specific day.

Step-by-step explanation:

Predicates in logic help clarify the precise meaning of statements within a specific domain. When considering a set of employees at a company, we can use predicates to describe various properties or actions related to these employees. The given predicates are:

• S(x): x was sick yesterday
• W(x): x went to work yesterday
• V(x): x was on vacation yesterday

The translations of the English statements into logical expressions are as follows:

1. Φ√ (S(x) ∧ W(x)) for 'Everyone was well and went to work yesterday.'
2. Φ√ (S(x) → ¬W(x)) for 'Everyone who was sick yesterday did not go to work.'
3. ∃x (S(x) ∧ W(x)) for 'Yesterday someone was sick and went to work.'
4. ∃x (¬S(x) ∧ ¬W(x) ∧ ¬V(x)) for 'Someone who missed work was neither sick nor on vacation.'
5. S(Ingrid) ∧ W(Ingrid) for 'Ingrid was sick yesterday but she went to work anyway.'

Answer 2

Step-by-step explanation:

Define Sets:

S(x): x was sick yesterday

W(x): x went to work yesterday

V(x): x was on vacation yesterday

Find:

Logical expression with the same meaning of:

(a) Everyone was well and went to work yesterday.

(b) Everyone who was sick yesterday did not go to work.

(c) Yesterday someone was sick and went to work.

(d) Someone who missed work was neither sick nor on vacation

(e) Ingrid was sick yesterday but she went to work anyway.

Solution:

a) Everyone was well and went to work. People who could have possibly gone to work would have been who were well or un-well so the the resulting set would be.

A (x) = W(x) - (W(x) & S(x))

b) Everyone who was sick and did not go to work. Sick people may or may not go to work. Hence, the domain can be defined as:

B (x) = S(x) - (W(x) & S(x))

c) Someone who was sick and did not go to work. Sick people may or may not go to work. Hence, the domain can be defined as:

C (x) ⊆ (W(x) & S(x))

d) Someone who missed work was not sick not on vacation:

D (x) = ( W'(x) & S'(x) & V'(x))

e) Ingrid was sick but went to work,, answer to part c and e are the same

E (x) = C(x) ⊆ (W(x) & S(x))