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Discrete Mathematics and Its Applications ( 8th International Edition ) ISBN:9781260091991
- 5. Use a table to express the values of each of these Boolean functions. а) F(x, у, 2) %3DХу b) F(x, y, 2) —х + уz с) F(x, у, 2) —ху+ (хуг) d) F(x, y, 2) — x(уz + у2)arrow_forward2. Recall that the Fibonacci sequence a₁, a2, a3,... is defined by a₁ = a₂ = 1 and An = An-1 + An-2 for all n ≥ 3. In this exercise, we will use determinants to prove the Cassini identity an+1ªn-1-a² = (−1)n for all n ≥ 2. Define suitable values for ao and a_1 so that the relation an = an−1 + An−2 holds for all n ≥ 1. (b) Let A = 01 (11) Show that an+k an+k+1= for all k-1 and all n ≥ 0. (c) Use (b) to show that An An+1 An-1 :) = = Then take the determinant on both sides to deduce the Cassini identity. = An An ak Ak+1 An ao a-1 a1 aoarrow_forward{u1, u2, u3} is an orthonormal set and x = c1u1 + c2u2 + c3u3. ||x|| = 5, <u1,x> = 4 , x ㅗ u2 , find c1,c2,c3arrow_forward
- a. Construct circuits from inverters, AND gates, and OR gates to produce the output (x + yz)(xy + z). b. Write sum-of-products expansion of the Boolean function F (x, y, z) = 0 if and only if ỹz = 0.arrow_forward6. Use a table to express the values of each of these Boolean functions. а) F(x, у, 2) —7 b) F(x, у, 2) %3D Ху+ Yz с) F(x, у, 2) — хӱг + (хуг) d) F(x, у, 2) 3 У(xz + X)arrow_forward2. Use a table to express the values of each of these Boolean function. F(x, y, z)= x+yzarrow_forward
- 6. Prove the following three Boolean Equations true (list the law you used for each step on your way to the proof): a. A+ A= A•(A+ B) + A• (B+ A) b. C®D=C•(C+ D) +C+D+E•D c. F•(G+H) = F •G+(F +G) + (F• G+ H) • (H + F)arrow_forwardLet B denote a Boolean algebra. Prove the identity V a, b e B, (a · b = 0) ^ (a + b = 1) = a = b. That is, prove the complement b is the unique element of B which satisfies (b · b = 0)^ (b + b = 1).arrow_forwardShow that U(20) + (k) for any k in U(20).arrow_forward
- 2. Let meN and a € Z. (a) If ged(a,m) = 1, then Bézout's lemma gives the existence of integersz and y such that ax + my = 1. Prove that a+mZ is the multiplicative inverse of a +mZ. (b) Determine the least nonnegative integer representative for (11+163Z)-¹ by expressing 1 as a linear combination of 11 and 163 (using the extended Euclidean algorithm).arrow_forwardLet M-{1, 2}. Then O IP(MXM)I=4 None of the mentioned MxM=(1, 4) M×M={(1,1),(2,2)} O IP(MxM)|=16arrow_forwardCompute the determinant for every n € N. 1 -(**) 3 2 det 2 n n+1 n-arrow_forward
- Linear Algebra: A Modern IntroductionAlgebraISBN:9781285463247Author:David PoolePublisher:Cengage Learning