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HIGHER COLLLEGE OF TECHNOLOGY

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1. Barry wentfor a swim = P

Mary went for a swim = Q

Q.P is similar to P.Q. They

In this sentence, which is not conditional, there are two propositions P( barrt went for

a swim) and Q ( Mary wentfor a swim). They are set up in an indipedent manner and

the key words is ‘and’. Both statements are antecedents.

(P ∧ Q) → (M ∨C) and (P ∧ H) ↔ (M ∨C)

2. In the sentences provided which is conditional, there are two prepositions A ( The

frogs are dancing under the bridge and B(Privided that either the moon is out or the

bats aren’t about). The relationship between the two propositions is depedent and in

this case the key words are ‘ Privided that’. It illustrates that on eis antecedent (A) and

the other is the consequent( B). The antecedent is the firts part of the logical

proposition.

3. The relationship established is disjunction, which means that it is false if both M and

B are false. It can be set to be true if B is true , if M is true or if M and B are true. It

can also be identified to have negotion or the logical compliment which usually takes

the preposition Ᵽ to a preposition which is not P. It is often interprated as truth when ꜚⱣ

(ꜚⱣ). The argument is Unirary logical connection.

1.

4.

J

S

J & S

(J&S →

B)

C ⊢ ~C

(J→S)

Ᵽ

ꜚⱣ

T

F

F

T

3

T

T

F

F

T

F

T

F

T

F

T

T

F

T

T

T

T

T

F

F

T

F

T

T

Tautology: for all probabilities of A and B, the statement is valid.

5.

P→B

(C →

~L)

(C → ~L) → (P

&~B)

(P & ~B) ⊢ L & C

T

T

F

F

T

F

T

F

F

F

T

T

T

T

F

F

The function from the proposition (T,F) entails a tautology and if the final interpretation P

satisfies the formula B then it is T │» BB is a valid tautology.

6. The statements are dependent and the whole sentence has four propositons where two of

each pair are depedent. I ( if the door is open when everyone is away) G (the coons will get

in). Second pair of the prepositions entails T ( Either the door is open) Y ( or everyone is

away) K ( the coons will get in ). The key words are If, either and so.

(I ∨G)

(I→ T)

(I→ G)∧(T →

I)

4

T

F

T

F

F

F

T

T

T

F

T

F

(q ∨r ) → p, (q → p)∧(r → p)

7. (E v D) & L, D→L ⊢ E

E V D

(¬L ∨ E) ∧ (¬L & ¬D)

E ∧ D

L ⊢ E D

8. (G & ~L) → (R v S), ~(G→L) ⊢ ~S→R

G & /~L

G » L

¬R ∨ S

R

~(G→L)

R

S

9. In the sentences provided which is conditional, there are two prepositions the goats been

hungry (A) and going to work (B).

(A → ((B → A) → A)) → (1) ((A → (B → A)) → (A → A)) by S A → ((B → A) → A) by K

(2) (A → (B → A)) → (A → A) by MP, (1), (2) (3) A → (B → A) by K (4) A → A by MP,

(3), (4)

A, B⇒ A A ∧ B⇒ A (∧l) ⇒ (A ∧ B) → A (→r) A, B⇒ B, A A⇒ B, B → A (→r) ⇒ A → B,

B → A (→r) ⇒ (A → B) ∨ (B → A) (∨r).

10. (Åx)(Fx£~Gx)

∀x(¬(P(x)∧H(x))

∀x (H(x) -> ~p(x))

5

11. Ax[S(x) --> L(x)]

S(x) = x some bar

L(x) = x certainity that is open

12. ~Ex[~B(x) & (W(x) & ~M(x))]

B(x) = x bob

W(x) = x sweet

M(x) = x not working

13. (x∀)(∀y)(Ayx → Bxy)

(x∀)(∀y)(Ayx) = Mary ownership of the cat

BXY= cat belonging.

15. ∀x)(∃y)Lxy it illustrates that everyone has someone he or she likes and not necessarily

same person. (∃y)(∀x)Lxy simplifies that a particular individual is loved by eceryone.

(∀x)(∃y)Lyx means that everyone is loved by someone and (∃y)(∀x)Lyx means that a

particular individual loves everyone.

16. →∀x(Cx → ~Wx)

∃y(Wy & Sy)

├ ~∃z(Sz & ~Cz)

│˄cx

wx

sy

17. ∀x∀y(Ey → Jx), ∀x(∀yEy v Lx) ├ ∀x(Jx v Lx)

∀x∀y(Ey → Jx),

(∀yEy & LX)

lx » jx

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¬ey ∨ ey

jx

~(G→L)

ey ∀yEy

18. ¬p ∧ ¬q ◦ ¬(p ∧ q) ∧L ¬p, ¬q ◦ ¬(p ∧ q) ¬L ¬R ¬q, p ∧ q ◦ p ∧L ¬q, p, q • p ¬(p ∧ q) ◦ ¬p

∧ ¬q ¬L ◦ p ∧ q, ¬p ∧ ¬q ∧R ◦ p, ¬p ∧ ¬q ◦ q, ¬p ∧ ¬q ∧R ◦ q, ¬p ¬R p

19. The relationship established is disjunction, , if M is true or if M and B are true. It takes the

preposition Ᵽ to a preposition which is not P. It is often interprated as truth when ꜚⱣ (ꜚⱣ). The

argument is Unirary logical connection.

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