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One important technique used to prove that certain sets not regular is the pumping lemma. The pumping lemma states that if
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Discrete Mathematics and Its Applications ( 8th International Edition ) ISBN:9781260091991
- Consider f : {1, 2, 3} → {a, b, c} and g : {a, b, c} → {apple, ball, cat}defined as f(1) = a, f(2) = b, f(3) = c, g(a) = apple, g(b) = ball and g(c) = cat.Show that f, g and gof are invertible. Find out f -1 , g-1 and (gof)-1 and show that (gof)-1= f –1o g-1arrow_forwardIf Vo and V; are respectively the t-conorms that are dual to the t-norms Ao and Ag respectively. Show that for any t-conorm V.arrow_forwardSuppose an additional binary-valued variable k bears the following relationship: i) Pr(X1,X6|X5, X7)=Pr(X1|X5, X7)Pr(X6|X5,X7) ii) Pr(X2,X3|X7)≠Pr(X2|X7)Pr(X3|X7) Incorporate the variable k into the Bayesian network in figure 1. Hint: you need to add into figure 1 by drawing the directional link(s) to/from the variable X7 from/to the relevant variable(s), and add the appropriate joint/conditional probability terms. However, you do not need to specify the exact value for the probability termsarrow_forward
- 2. Let V be the set of all ordered pairs of real numbers, and consider the following addition and scalar multiplication op- erations on u = (u₁, U₂) and v = (V₁, V₂): u + v = (u₁ + v₁ + 1, U₂ + v₂+1), ku= (ku₁, ku₂) (d) Show that Axiom 5 holds by producing an ordered pair -u such that u + (-u) = 0 for u = (u₁, U₂). (e) Find two vector space axioms that fail to hold. och equipped witharrow_forwarda ) Let R be the relation on the set A={1,2,3,4} defined by aRb if and only if 2a>b+1. Find the matrix representing R∘R. b) Suppose that the relation R is defined on the set Z where aRb means a= ±b. Show that R is an equivalence relation.i would like to get a non handwriting answer to be easy to cope pleasearrow_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
- Suppose A = {a1, . . . , am} and B = {b1, . . . , bn}. Let R and T be relations fromA to B with matrices M and N respectively. How is the matrix of R ∪ T obtained from Mand N. How about the matrix of R ∩ T.Suppose C is a non-empty set and that T is a relation from B to C. We define the compositerelation R ◦ T from A to C by declaring aT c if there is b ∈ B such that aRb and bRcarrow_forwardb) Check whether the relation R on the set S = {1, 2, 3} with the the matrix 1 1 1 0 1 1 1 1 1 is an equivalence relation. Which of the following properties R has: reflexive, symmetric, anti-symmetric, transitive? Justify your answer in each case?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_forward
- 1) Let set A = {1,2,3,4} and let R1 and R2 be binary relations on A. Specifically, let: R1 = {(1, 1), (1, 2), (2, 1), (2, 2), (2, 4), (3, 4), (4, 2), (4, 3) (4, 4)} R2 = {(1, 2), (1, 3), (1, 4), (2, 1), (2, 3), (4, 1), (4, 2)} Determine the following: a) Whether R1 is reflexive, irreflexive, symmetric, anti-symmetric and/or transitive. b) Whether R2 is reflexive, irreflexive, symmetric, anti-symmetric and/or transitive. c) R1 o R2. d) R2 • R1. e) R1 U R2. %3D f) R1 n R2. g) The reflexive, symmetric and transitive closures of both R1 and R2.arrow_forwardIf H is a Hilbert space and A € BL (H) is positive, prove that A+21 is invertible.arrow_forwardProve that the following relations are true in general: A₂=(A₁-A₂) U (A₂-A₁) a. A₁ + b. A₁ U (A₂ MA3)=(A₁ UA₂) N (A₁ UA3)arrow_forward
- Linear Algebra: A Modern IntroductionAlgebraISBN:9781285463247Author:David PoolePublisher:Cengage Learning