First Order Logic doubt

Question

Consider the formula:∀x(P(x)→Q(x)) ∧ ∀x(Q(x)→R(x)) → ∀x(P(x)→R(x))

Which is true?

A. Tautology
B. Contingent
C. Contradiction
D. Unsatisfiable

Answer 1

Try this format for easier solution

$$((p\to q)\wedge (q\to r))\to (p\to r)$$

 

$$now\; to\; check\; tautology\; ,\; we\; just\; need\; to\; make\; the\; given\; equation\; false\; for\; 1\; case$$

$$that\; is\; possible\; only\; if\; LHS\; of\; implication\; is\; TRUE\; and\; RHS\; of\; implication\; is\; FALSE$$

 

$$Lets\; make\; RHS\; True\; first$$

so take p = True and r = false

$$so\; p->r\; becomes\; false$$

 

now for q = true and q= false , check the whole implication

 

for q = true

 

$$((p\to q)\wedge (q\to r))\to (p\to r)$$

((T$$\to$$T)$$\wedge$$(T$$\to$$F)) $$\to$$(T$$\to$$F)

= T

 

for q = false

 

$$((p\to q)\wedge (q\to r))\to (p\to r)$$

((T$$\to$$F)$$\wedge$$(F$$\to$$F)) $$\to$$(T$$\to$$F)

= T

 

So we can say its a Tautology

 

similarly to check Contradiction , we can check for 1 case where given equation is TRUE 

@GO Classes i hope i am using the correct logic