3. Given F(A,B,C,D,E) = TM (0,3,6,9,11,19,20,24,25,26,27,28,20,30). TTD (1,2,12,13). Find two minimum SOP expansions for F. Underline the essential prime implicants in your answer.
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- 1. Complete the implicant table to find all the prime implicants for the given minterm expansion F(a, b, c, d) = Em(0, 1, 2, 5, 6, 9, 12) 2. Now using the prime implicants from problem 1, construct a PI chart and find out all the essential prime implicants. Also, do you need any additional prime implicant, if so, draw a reduced PI chart and then finally determine the minimum SOP for problem 1.Consider the minterm expansion for f=Σm(1, 3, 4, 5, 6, 7, 9, 11) with 4 variables. (a) Draw its Karnaugh map. (b) Write down expression for all the prime implicants. (c) Write down expression for all the essential prime implicants. (d) Write down the minimum sum of product expression for f.Explain what is Euler number o(m) for a natural number m > 1. Give a proof of Euler Theorem, whose proof you could write down straightforward by generalising that of Fermat little Theorem. (cf. the handout on Moodle Topic 12).
- prove that F ty Simplify the following Boolean function using sum-of-products format F(A.B.C.D) = E (0, 2, 3, 7, 8. 10. 11, 12, 13, 14, 15)For 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.Find GCD(270,192) by applying Euclid’s algorithm.
- F(w, x, y, z)=Σ m(0,1,2,8,9,10) + X(4,6,12,14) Karnaugh Map and give the simplified logical expression derived from Karnaugh Map with indicating every simplification step below.Given the Boolean function F(A, B, C, D) =(0, 1, 6, 7, 10, 11) together with the don't care conditions d (A, B,C,D)= (2,3,4,5,8,9,14,15). a) Simplify F in sum of products (SOP). b) Simplify F in product of sums (POS). c)implement the function f in bothWe can analyse cyclic codes using the fact that they correspond to certain subsets of the ring F,[r]/(x-1). What type of subsets correspond to cyclic codes? What property of those subsets corresponds to the cyclic condition, and why? What characteristic of the code does the parameter n in the ring correspond to?
- Prove, by finding constants C₁, C₂, and no that satisfy the definition of order of magnitude, that f = (g) if f(x) = 3x³ - 7x and g(x) = x³12.Using K-maps, simplify the following Boolean functions and express each of them in sum-of- products (SOP) and product-of-sums (POS) forms. a) F(w, x, y, z) = Σ(0, 1, 2, 5, 8, 10, 13) Σ(1, 3, 6, 7, 8, 9, 12, 13, 14, 15) Σ(2, 4, 10, 12, 14) and d (w, x, y, z) = (0,1,5,8) (0, 4, 8, 9, 10, 11, 12, 14) b) F(w, x, y, z) = c) F(w, x, y, z) = d) F(w, x, y, z) =6. Show that if a, b, c, and mare integers such that m≥2, c > 0, and a≡b (mod m), then ac ≡bc (mod mc).