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DISCRETE MATHEMATICS LOOSELEAF W/CONNECT
- Represent the following relation {1,2, 3, 4} with a matrix. {(1,1), (1, 4), (2, 2), (3, 3), (4, 1)}arrow_forwardLet X = {1,2,3,4}, and consider binary relations R and S, both subsets of X X X, defined defined as follows. (i) R = {(1,1), (1,2), (2,1), (2,2), (4,1)} and (ii) S= R_n{(i,t)\i e X} 1. Precisely list the elements of R-1. Is R a function from X to X (Yes/No and Justify)? How many elements does SX R contain? (Equivalently, find |S X R.) Justify your answer.)arrow_forward26 of 40 Consider the Datalog programs P1 (left) and P2 (right) below, which use relations R(A, B) and S(A, B). P1 P2: T1(A) R(A, B). T4(A) R(A, B), S(A, B). T2(A) S(A, B). T3(A) + T1(A), T2(A). Which of the following statements is TRUE about the relationships between relations T3 and T4 defined by P1 and P2, respectively? Note that the commas "," used in the rule bodies to separate the predicates is the same as using AND. Select one: T3 and T4 include the same set of tuples. Every tuple in T3 is also contained in T4, that is, T3 C T4. O None of the other answers, that is, T3 and T4 contain different tuples, in general. O Every tuple in T4 is also contained in T3, that is, T4 C T3.arrow_forward
- 2. 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_forwardEvaluate each determinant in Exercises 1-10. |5 1. 2 3 4 8 2. 5 6 7 -4 1 7 3. 5 6 4. -2 -5 -7 5. 14 1 -3 6. -8- -4 2 -5 7. 8. -2 9. 10. on-len m -in o lem- 2.arrow_forwardA model can have both deterministic and random inputs in different components; which elements are modeled as deterministic and which as random are issues of modeling realism. Select one: True Falsearrow_forward
- A binary operator f on a set X is commutative if f(x, y) = f(y, x)for all x, y ∈ X. State whether the givenfunction f is a binary operator on the set X. If f is not a binaryoperator, state why. State whether or not each binary operator iscommutative. f(x, y) = x ∪ y, X = P({1, 2, 3, 4})arrow_forward5) Let R be a binary relation over the positive integers defined as follows: R = {(a, b) | 2b < a < 3b } Determine whether R satisfies the following properties. Give brief justifications for your answers. (i) (i) (ii) (iv) (v) reflexive irreflexive symmetric anti-symmetric transitivearrow_forwardFor the binary relation R (defined below) on A and each tuple (a, b) indicating that a is related to b, which of the following statements would be true? R={ (0,4), (1, 1), (1, 3), (1, 4), (2, 1), (2, 2), (3, 2), (4, 4) } where A = {0, 1,2, 3, 4} Select one: O Ris symmetric but not antisymmetric O Ris antisymmetric but not symmetric O Ris both symmetric and antisymmetric O none of these options O Ris neither symmetric nor antisymmetricarrow_forward
- 2. Let U = {1, 2, 3} and A = U × U. In each case, show that = is an equivalence on A and find the quotient set A. (a) (a, b) = (a₁, b₁) if a + b = a₁ + b₁. (b) (a,b) = (a₁, b₁) if ab = a₁b₁. (c) (a, b) = (a₁, b₁) if a = a₁. (d) (a, b) = (a₁, b₁) if a - b = a₁-b₁.arrow_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_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_forward
- Linear Algebra: A Modern IntroductionAlgebraISBN:9781285463247Author:David PoolePublisher:Cengage LearningElementary Linear Algebra (MindTap Course List)AlgebraISBN:9781305658004Author:Ron LarsonPublisher:Cengage Learning