Question 2: Let f(x):R¬R, f(x) = 2x² +5. a. Is f(x) one-to-one? Prove your answer. b. Is f(x) onto? Prove your answer. c. Is f(x) bijection? Prove your answer.
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A: The answer is given in the below step
Q: Question 3 (a) Show that [¬p v (p ^ q)] ^ ¬q → ¬(p v q) by using logical equivalence identities.
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A: Lets see the solution in the next steps
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A: Solution - In the given question, we have to find whether the given statement is a tautology or not.
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A: Solution -
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Q: q ∧ r) are logically equivale
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Q: Show that (p ∧ q) → r and (p → r) ∧ (q → r) is logically equivalent.
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A: Solution:
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- 3. Consider the formula A=Vx (p(x) V q(x)) → (Vxp(x) Vrq(x)). (a) Show that A is valid. (b) Show that the converse of A is not valid.We have learned the mid-point and trapezoidal rule for numercial intergration in the tutorials. Now you are asked to implement the Simpson rule, where we approximate the integration of a non-linear curve using piecewise quadratic functions. Assume f(x) is continuous over [a, b] . Let [a, b] be divided into N subintervals, each of length Ax, with endpoints at P = x0, x1, X2, ..., Xn,..., XN. Each interval is Ax = (b – a)/N. The Simpon numerical integration rule is derived as: N-2 Li f(x)dx = * f(x0) + 4 (2n odd f(xn)) + 2 ( En=2,n even N-1 f(x,) + f(xn)] . Now complete the Python function InterageSimpson(N, a, b) below to implement this Simpson rule using the above equation. The function to be intergrate is f (x) = 2x³ (Already defined, don't change it). In [ ]: # Complete the function given the variables N,a,b and return the value as "TotalArea". # Don't change the predefined content, only fill your code in the region "YOUR CODE" from math import * def InterageSimpson (N, a, b): # n is…We have learned the mid-point and trapezoidal rule for numercial intergration in the tutorials. Now you are asked to implement the Simpson rule, where we approximate the integration of a non-linear curve using piecewise quadratic functions. Assume f(x) is continuous over [a, b] . Let [a, b] be divided into N subintervals, each of length Ax, with endpoints at P = x0, x1, x2,.. Xn,..., XN. Each interval is Ax = (b − a)/N. The equation for the Simpson numerical integration rule is derived as: f f(x) dx N-1 Ax [ƒ(x0) + 4 (Σ1,n odd f(xn)) ƒ(x₂)) + f(xx)]. N-2 + 2 (n=2,n even Now complete the Python function InterageSimpson (N, a, b) below to implement this Simpson rule using the above equation. The function to be intergrate is ƒ(x) = 2x³ (Already defined in the function, no need to change).
- We have learned the mid-point and trapezoidal rule for numercial intergration in the tutorials. Now you are asked to implement the Simpson rule, where we approximate the integration of a non-linear curve using piecewise quadratic functions. Assume f(x) is continuous over [a, b]. Let [a, b] be divided into N subintervals, each of length Ax, with endpoints at P = x0, x1,x2,..., X., XN. Each interval is Ax = (b − a)/N. The equation for the Simpson numerical integration rule is derived as: f f(x)dx ≈ [ƒ(x0) + 4 (EN-1,n odd S(x)) + 2 (Σ2²n even f(x)) + f(XN)]. Now complete the Python function InterageSimpson (N, a, b) below to implement this Simpson rule using the above equation. The function to be intergrate is f(x) = 2x³ (Already defined in the function, no need to change). *Complete the function given the variables N, a,b and return the value as "TotalArea"." "Don't change the predefined content' only fill your code in the region *YOUR CODE"" from math import * def InterageSimpson (N, a,…2. For the given Boolean function: F (A,B,C,D,E) = X(0, 2,3,4,5,6,7,11,15,16,18,19, 23,27,31) Show the area enclosed and label it with the corresponding literal, legibly. Obtain the simplest function in sum of product form and product of sum form using map methodFor the function f(w,x,y,z) = E (m4, m7, m8, m9, m10, m12, mi13, m15), complete the parts of this question below. (a) Draw and fill in a correct, properly-labelled Karnaugh map for f(w,x,y,z). (b) List ALL sum-of-prod. prime implicants, and identify which ones are essential. (c) Write the complete least-cost sum-of-products expression for f(w,x,y,z). (d) Now consider m2, m5, and m11 as don't-care cases for f(w,x,y,z) in part (a). Provide the least-cost sum-of-products expression using this information.
- Determine whether the function f(x) = 4x−1 where f:R→R is a bijection. If it is not, explain why not.Can someone please explain the answer to the minimum expression of the x(a,b,c) = [ (0,3,5,6) using Karnaugh map? For this function, I am unable to form groupings due to the positions of 0, 3, 5, and 6 (diagonal?). So, I am kind of confused. Hoping to received detailed explanation.Simplify the following Boolean expressions using four-variable maps: F (W, X, y, z) = I (1,4,5,6,12,14,15) 1. For the Boolean function F given in the truth table, find the following: (a) List the minterms of the function. (b) List the minterms of F. (c) Express Fin sum of minterms in algebraic form. (d) Simplify the function to an expression with a minimum number of literals.
- Simplify the following Boolean functions, using K-maps: F (A, B, C, D)=Σ(3, 7, 11, 13, 14, 15)For f(a, b) = (a | b) | b (a) Simplify f(a, b). (b) Find DNF for f(a, b). (c) Is f(a, b) satisfiable?2 Simplify the following Boolean function, using four variables K-map. F(A, B, C, D) = E(2, 3, 6, 7, 12, 13, 14)