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DISCRETE MATHEMATICS LOOSELEAF
- Please just answer parts iv. and v.arrow_forwardDetermine the Determinants of mat A 1 -2 3 mat A = 8 7 |-3 1 -9 -10]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_forward
- There are 16 binary relations on the set {0, 1}: (a) { } (d) {(1,0)} (g) {(0, 0), (1, 0)} (j) {(0, 1), (1, 1)} (m) {(0,0), (0, 1), (1, 1)} (b) {(0, 0)} (e) {(1, 1)} (h) {(0, 0), (1, 1)} (k) {(1, 0), (1, 1)} (c) {(0, 1)} (f) {(0, 0), (0, 1)} (i) {(0, 1), (1,0)} (1) {(0,0), (0, 1), (1, 0)} (0) {(0, 1), (1, 0), (1, 1)} (n) {(0,0), (1,0), (1, 1)} (p) {(0, 0), (0, 1), (1, 0), (1, 1)} For a-f use the letters for each relation above. a. List the reflexive relations on the set {0, 1}. b. List the irreflexive relations on the set {0, 1}. c. List the symmetric relations on the set {0, 1}. d. List the transitive relations on the set {0, 1}. e. List the antisymmetric relations on the set {0, 1}. f. List the asymmetric relations on the set {0, 1}.arrow_forward3. Show that any language A is recognizable if and only if Aarrow_forwardSuppose I + AB is invertible. Show that I + BA is also invertible by showing that (I + BA)- =I – B(I + AB)-A.arrow_forward2. 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 the domain of x and y be all people. Let W(x): "x is a woman". M(y): "y is a man", and L(x.y): "y loves x". Then vxay. (W(x)AM(y))L(xr, y)] means Every man has a woman that loves him. Every man is loved by all women. None of these Every woman is loved by all men. Every woman has a man that loves her.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_forwardLet U = the set of all trees M(x) = "x is a Maple tree", E(x) = "x is an evergreen tree" %3D %3D Select all of the statements that are logically equivalent to: "It is not the case that for some tree, being an evergreen implies being a maple." (Ex E U, E(x) = M(x)) (Ax € U, E(x) → M(x)) Væ E U,¬ (E(x) = M(x)) Ξ0ευ,- (E(π) → M(x)) Vx E U, E(x) ^ ¬M(x) Jæ E U, E(x) ^ ¬M(x) Væ E U,¬M(x) ¬E(x) 3x E U, ¬M(x) → ¬E(x)arrow_forwardEXERCISE 1.13.3: Show an argument with quantified statements is invalid. Show that the given argument is invalid by giving values for the predicates P and Q over the domain (a, b). (a) x (P(x) → Q(x)) 3x - P(x) :: 3x -Q(x) 3x (P(x) v Q(x)) 3x -Q(x) :: 3x P(x) ~ Feedback?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_forwardarrow_back_iosSEE MORE QUESTIONSarrow_forward_ios
- Linear Algebra: A Modern IntroductionAlgebraISBN:9781285463247Author:David PoolePublisher:Cengage Learning