Prove by natural deduction: Show ∃x[p(x) ∧ q(x)] ⊧ ∃xp(x) ∧ ∃xq(x)

Q: Give a proof by cases that x ≤ |x| for all real numbers x.

A:

Q: For f(a, b) = (a | b) | b (a) Simplify f(a, b). (b) Find DNF for f(a, b). (c) Is f(a, b)…

A: (a) f(a, b) = (a | b) | b can be simplified as follows: f(a, b) = (a | b) | b = b | b (since a | b =…

Q: Prove valid. (x) ( ~ Ax > Dx)                                                  / (Ex) (Ax v Bx) (Ey)…

A: Below is the answer to above question. I hope this will helpful for you..

Q: Prove the argument(∀x)[(B(x) ∨ C(x)) → A(x)] → (∀x)[B(x) → A(x)]

A: Given argument, (∀x)​[(B(x) V C(x)) → A(x)] → (∀x)​[B(x) → A(x)]

Q: Prove or disprove that for any x ∈ N, x(x+1)/2 ∈ N (where N = {0, 1, 2, 3, ….}

A: Hello student

Q: Prove that f(x) = x is O(x3).

A: Attached image below:

Q: Show that (p ∧ q) ∨ ( p ∧ q) ≡ p using rules.

A: To show that (p ∧ q) ∨ ( p ∧ q) ≡ p using rules. Let us write ^ as '*' sign and other sign as '+'.…

Q: Simplify the following Boolean functions, using K-maps: F (A, B, C, D)=?(1,3,6,9,11,12,14)

A: The given equation is: F (A, B, C, D)=?(1,3,6,9,11,12,14) we'll use sum f products to minimize or…

Q: Determine P(A x B) – (A x B) where A = {a} and B = {1, 2}.

A: Given sets: A = {a} and B = {1, 2} Now  A × B= {(a, 1), (a, 2)}

Q: Suppose that f(x) is O(g(x)).   Does it follow that 2f(x) is O(2g(x))?   Prove your answer.

A: The given information is: f(x) is O(g(x)).

Q: Use the Pumping Lemma with length to prove that the following language is non-regular: L = {a²ba",…

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Q: 10 Is A-(BU C) = (A-B) (A-C)? Prove it or disprove it.

A:

Q: Prove : ?3 − 4? + 6 is divisible by 3 for all ? ∈ ℕ

A: According to the information given:- We have to prove the given equation.

Q: For each of the following recurrences, give an expression for the runtime T (n) if the recurrence…

A:

Q: For f (a, b) = (a | b) | b (a) Simplify f (a, b). (b) Find DNF for f (a, b). (c) Is f (a, b)…

A: In addressing the function f(a,b)=(a∣b)∣b, we simplify it to f(a,b)=a∣b, leveraging the…

Q: It was claimed that: (a, b) ≤ (c, d) ⇔ (a < c) ∨ (a = c ∧ b ≤ d) defines a well-ordering on N x N.…

A: A well-ordering is a total order on a set where every non-empty subset has at least element. It must…

Q: L = {w ∈ {a, b}∗ | the length of w is a multiple of 3 and w contains more a’s than b’s}. Use…

A: We need to demonstrate that there are infinitely many equivalence classes under the…

Q: Show validity by natural deduction: ∀x[p(x) → q(x)] → [(∃xp(x) ∧ ∃xr(x)) → ∃xq(x)]

A: m trying to follow my textbook but don't have many examples likes this one. Any help would be…

Q: Make a truth table for the following expression [8]: (a) ¬[(¬p −→ q) ∨ (¬p ∧ ¬q)] (b) Based on…

A: Solution:   Given,           ¬[(¬p −→ q) ∨ (¬p ∧ ¬q)]

Q: Prove the following argument is valid: Væ [p(x) → (q(x) V r(x)] Væ (r(x) → q(x)] -3x q(x) 3x p(x) ..

A: We use Universal Generalization, Universal Specification, and Existential Generalization and…

Q: 7- Prove that (A+B)(A+C)=A+BC.

A: This question comes from Digital Electronics which is a paper of computer science. Let's discuss it…

Q: If f(n) = O(g(n)), then g(n) = Ω(f(n))

A: Given : - The functions f(n) and g(n) be functions with domain {1, 2, 3, . . .} Asymptotic analysis…

Q: Formally prove that on? + 10n + 10 = 2(n²) using the definition of N(·).

A: Proved the given function

Q: et G: (0,1)" (0,1)"+l be defined as follows: (X1. ....x,) = (X, x, OX2, X OX2 O X3,X x2 ... x, X).…

A:

Q: Let A = {(N1, N2) | N1 and N2 are NFAS and L(N1)N L(N2) = Ø}. Show that A is %3D decidable.

A: Given that, A={ &lt;N1, N2&gt; | N1 and N2 are NFAs and L(N1)∩L(N2)=∅} That means N1 and N2 are…

Q: ¬p → ​(p ∧ q​)  truth table

A: First of all,