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Discrete Mathematics and Its Applications ( 8th International Edition ) ISBN:9781260091991
- a. Let R be the equivalence relation defined on Z in Example 2, and write out the elements of the equivalence class [ 3 ]. b. Let R be the equivalence relation congruence modulo 4 that is defined on Z in Example 4. For this R, list five members of equivalence class [ 7 ].arrow_forward5. Using the following context free grammar give a parse tree for the strings. E → E +T|T T → TxF|F F → (E)\a (a) a+a+a (b) ((a))arrow_forwardFor non empty binary relation R={(a, a),(a, b),(a, e),(b, b),(b, e),(c, c),(c, d),(d, d).(e, e)} on the set A={a, b, c, d, e}, which is the following is true? O Reflexive, Anti-Symmetric O Reflexive O Reflexive, Symmetric, Transitive O Reflexive, Transitive O Reflexive, Symmetric, Anti-Symmetric, Transitive O Reflexive, Anti-Symmetric, Transitive O Reflexive, Symmetric O Transitive O Symmetric O Symmetric, Transitive O Anti-Symmetric, Transitivearrow_forward
- In the Axiom Systems section of the Background chapter (1.6), a binary relationarrow_forwardLabel the following statement as True (T) or False (F): Let L be any Turing recognizable language. Then L is Turing recognizable.arrow_forward2. Answer each part for the following context-free grammar. R→ XRX | S S-> aTb | bТа T→ XTX|X | € X→ a | b a. What are the variables and terminals of G? Which is the start symbol? b. Give three examples of strings in L(G). c. Give three examples of strings not in L(G).arrow_forwardFinite language is a language with finite number of strings in it, i.e., there exist exactly k strings in this language such that k eNand k #00. For a finite language L, let |L| denote the number of elements of L. For example, |{A, a, ababb}| = 3. (Do not mix up with the length |x| of a string x.) The statement |L,L2| = |L1||L2| says that the number of strings in the concatenation LL2 is the same as the product of the two numbers |L1| and |L2|. Is this always true? If so, prove, and if not, find two finite languages L1, L2 S {a, b}* such that |L1L2| # |Li||L2l.arrow_forwardShow that the number of equivalence relation in the set {1, 2, 3} containing(1, 2) and (2, 1) is two.arrow_forward1) What is the following recursive set S? • 3 ∈ S • x,y ∈ S → (x + y) ∈ S 2) Consider the following recursive set S ⊂Z×Z. Give an example of a pair of integers a,b that is not in the set S. • (3,5) ∈ S • (x,y) ∈ S → (x + 2,y) ∈ S • (x,y) ∈ S → (y,x) ∈ S • (x,y) ∈ S → (−x,y) ∈ Sarrow_forward2) Let K be any Boolean algebra. A useful relation × ≤ y (read as “x precedes y” ) if and only if xy=x. i) ii) iii) a) If K is the Boolean Algebra of subsets of a set S, to what familiar relation on the subsets of S does Correspond? Refer to example 7.1 in page 348 in the textbook b) Use the axioms and laws of Boolean algebra to prove the following properties of ← in an arbitrary Boolean algebra K. Make sure that when you use the axioms or laws, write that down in the proof x < x for all x € K (Reflexive property) If xy and y If x can be defined as the elements of K as follows: y and y x, then x=y (Antisymmetric property) z, then x z (Transitive property)arrow_forwardLet A = {1, 2, 3}. Then number of equivalence relations containing (1, 2) is. Chose the correct.(A) 1 (B) 2 (C) 3 (D) 4arrow_forwardTo show that the set of all finite strings over the alphabet {0,1,2} is countable, you would define a bijection f: {0,1,2} - N, where f(A)=0 (remember that A is the empty string), listing strings in string order (with 0<1<2). Under this bijection, the f(201) = Your Answer: Answer Suppose that JA|=29, |B|=20, and |A n B| = 10. What is |A U B|? Your Answer: Answerarrow_forwardarrow_back_iosSEE MORE QUESTIONSarrow_forward_ios
- Elements Of Modern AlgebraAlgebraISBN:9781285463230Author:Gilbert, Linda, JimmiePublisher:Cengage Learning,