GATE Computer Science Engineering(CSE) 2027 Test: Propositional & First

F(x) ==> x is my friend
P(x) ==> x is perfect

D is the correct answer.

A. There exist some friends which are not perfect

B. There are some people who are not my friend and are perfect

C. There exist some people who are not my friend and are not perfect.

D. There doesn't exist any person who is my friend and perfect

S2 is true, as X' + Y' = Y' + X'

S1 is false.
Let P = 1, Q = 1, R = 0, we get different results
(P # Q) # R = (P' + Q')' + R' = (0 + 0)' + 1 = 1 + 1 = 1
P # (Q # R) = P' + (Q' + R')' = 0 + (0 + 1)' = 0 + 0 = 0

(A) "There exist some numbers which are either real OR rational"
(B) "All real numbers are rational"
(C) "There exist some numbers which are both real AND rational"
(D) "There exist some numbers for which rational implies real" (See Propositional Logic for details)
Clearly answer C is correct among all

So the predicate is evaluated asP(x) = (¬(x=1))∧(∀y(∃z(x=y*z)⇒((y=x)∨(y=1))) P(x) being true means x ≠ 1 and For all y if there exists a z such that x = y*z then y must be x (i.e. z=1) or y must be 1 (i.e. z=x) It means that x have only two factors first is 1 and second is x itself. This predicate defines the prime number.

∀ x ∃ y ∃ t(¬ F(x,y,t)) => ∀ x ¬(∀ y ∀ t F(x,y,t))

=> This statement can be expressed as => For all X, x can be either gold or silver then the ornament X is precious => For all X, (G(X) v S(x)) => P(X).

P ↔ Q = (P → Q) ∧ (Q → P)
P ↔ Q = (~P ∨ Q) ∧ (~Q v P)
So, option (B) is correct.

According to negation property of universal qualifier and existential quantifier

Considering each option : 
(A) If everything is a FSA, then there exists an equivalent PDA for everything. 
(B) It is not the case that for all y if there exist a FSA then it has an equivalent PDA. 
(C) Everything is a FSA and has an equivalent PDA. 
(D) Everything is a PDA and has exist an equivalent FSA. 
Thus, option (A) is correct. 
 
Please comment below if you find anything wrong in the above post.

I and II are same by Demorgan's law The IIIrd can be simplified to I.

Option A and option C are same G(x)-->C(x) can also be written as ~G(x) or C(x), and they are the correct representation. Option C says :There exists a graph and graph is not connected ,which is equivalent to given sentence. Option D: Every x which is graph is not connected. Option D is correct choice

We can easily prove that for any formula, there is a truth assignment for which at least half the clauses evaluate to true . Proof : Consider an arbitrary truth assignment. For each of its clause ‘j’ , introduce a random variable. X_j = 1 if clause ‘j’ is satisfied X_j = 0 otherwise Then, X = summation of (j * X_j) is the number of satisfied clauses. Given any clause ’c’ , it is unsatisfied only if all of its ‘k’ constituent literals evaluates to false as they are joined by OR operator. Now, because each literal within a clause has a 1/2 chance of evaluating to true independently of any of the truth value of any of the other literals, the probability that they are all false is (1 / 2)^k . Thus, the probability that ‘c’ is satisfied = 1 − (1 / 2)^k So, E(X_j) = 1 * (1 / 2)^k = (1 / 2)^k Therefore, E(X_j) >= 1/2 Summation on both sides to get E(X). Therefore, we have E(X) = summation of (j * X_j) >= m/2 where ‘m’ is the number of clauses. E(X) represents expected number of satisfied clauses. Thus, there must exist an assignment that satisfies at least half of the clauses. Please comment below if you find anything wrong in the above post.

A formula is satisfiable if there is atleast one assignment for which it is true. Clearly this formula is satisfiable as there are 7 assignments for which it is true.
A formula is valid if it is true for all assignments, which is not the case here.
Thus, option (B) is correct.

(A) a -> b means if ‘a’ is true then ‘b’ is also true. Now, ‘b’ is true. Therefore, ‘c’ is also true. => Using transitivity rule , a -> c 
(B) a <-> c is true if both ‘a’ and ‘c’ have same values. Let ‘a’, ‘b’ and c’ be false. Expression 'a and b' is false and expression 'not b' is true. RHS of the given equation should be true. But it evaluates to be false. Therefore, contradiction is there. 
Option (B) is not a tautology. 
(C) Let ‘a’ and ‘c’ be false. ’b’ be true a and b and c is false c or a is false 
(D) Let ‘b’ be true. b -> a means if ‘b’ is true then ‘a’ is also true. Therefore, ‘a’ and ‘b’ both evaluates to be true. 

Thus, option (B) is correct. 
 
Please comment below if you find anything wrong in the above post.

(A) Note that (p ∧ ~q) ≡ ~(p -> q). So it means rainy day to cold implication is false for all days. Which means non-rainy days are cold. (B) For all days, if day is not rainy, then it is cold [Non-Rainy days are cold] (C) There exist some days for which not rainy implies cold. [Some non-rainy days are cold] (D) Note that (p ∧ ~q) ≡ ~(p -> q). So it means rainy day to cold implication is false for some days. Which means not all rainy days are cold.

The statement "Tigers and lions attack if they are hungry or threatened" means that if an animal is either tiger or lion, then if it is hungry or threatened, it will attack. So option (D) is correct. Don't get confused by "and" between tigers and lions in the statement. This "and" doesn't mean that we will write "tiger(x) ∧ lion(x) ", because that would have meant that an animal is both tiger and lion, which is not what we want.

Option D is correct.

For P1, the right-hand side (A → C) ∧ (B → C) is equivalent to (¬A ∨ C) ∧ (¬B ∨ C), which simplifies to ¬(A ∨ B) ∨ C. This is exactly (A ∨ B) → C.

Thus P1 asserts (A ∧ B) → C ≡ (A ∨ B) → C, which need not hold. For a counterexample take A = true, B = false, C = false. Then (A ∧ B) → C is true while (A ∨ B) → C is false. Hence P1 is not a tautology.

For P2, the right-hand side (A → C) ∨ (B → C) is equivalent to (¬A ∨ C) ∨ (¬B ∨ C) = ¬(A ∧ B) ∨ C, i.e. (A ∧ B) → C.

Thus P2 asserts (A ∨ B) → C ≡ (A ∧ B) → C, which also need not hold. Using the same assignment A = true, B = false, C = false, the left side (A ∨ B) → C is false while the right side (A ∧ B) → C is true. Hence P2 is not a tautology.

Since neither P1 nor P2 is always true, both are not tautologies. Therefore the correct answer is Option D.

Option D is correct; A ∧ B is equivalent to ~(~A □ B).

From the given truth table, the binary operator satisfies the Boolean identity X □ Y = X ∨ ~Y for arbitrary propositions X and Y.

Compute (~A □ B): using the identity this equals ~A ∨ ~B, which by De Morgan's law is ~(A ∧ B).

Negating that gives ~(~A □ B) = ~(~(A ∧ B)) = A ∧ B. Hence option D simplifies to A ∧ B.

Answer : [ B ]
X = (P ⋁ Q) → R
= ~(P ⋁ Q) ⋁ R

= (~P ⋀ ~Q) ⋁ R

= (~P ⋁ R) ⋀ (~Q ⋁ R)

= (P → R) ⋀ (Q → R)

X → Y is true as (A ⋀ B) → (A ⋁ B) is always TRUE but reverse implication is not always true.