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- a. The relation x ≡ y if and only if x mod 4 == y mod 4 is an equivalence relation. Use this equivalence relation to partition the set {2, 4, 5, 6, 9, 23, 24, 31, 37} into equivalence classes. b. How many substrings “aab” are there in wwRw, where w = aabbaab? c. Prove that (LR)* = (L*)R for all languages L.Give conditions under which the following expressions are equivalent:A,Bγagg(C)(E1 ⋈ E2) and (Aγagg(C)(E1)) ⋈ E2where agg denotes any aggregation operation. How can the above conditionsbe relaxed if agg is one of min or max?Given set A = {a, b, c, d} show the equivalence relation, which contains eight ordered pairs, that induces this partition of A : {{a, c}, {b, d}}.
- 2. Use set identities to prove the following equivalence: A (BA) = AQUESTION 9 Show by membership that for all sets A, B and C: A - (AnB) CA – BShow that the relation R consisting of all pairs (x, y) suchthat x and y are bit strings of length three or more thatagree except perhaps in their first three bits is an equivalence relation on the set of all bit strings of length threeor more.
- Computer Science Let L = {w ∈ {0, 1}*: w has an even number of 0s and the last character of w is a 1}. Give the equivalence classes of the relation ≡L using regular expressions.Consider an equivalence relation R on X={a,b,c}. R= {(a,a), (a,c), (b,b), (c,a), (c,c) } [a]=Use the standard process we saw in class to derive an NFA from the RE (a|b)∗ba, and then use the subset algorithm to give an equivalent DFA. Either formalizing the DFA/NFA or diagramming them clearly will be sufficient. The final DFA should be labled with sets of NFA states. Apply Hopcroft’s DFA Minimization algorithm to your DFA from problem 3. Show the partitions at each refinement and the final DFA. Diagramming these DFAs is sufficient.
- For non empty binary relation R={(a, a),(a, b),(b, a),(b, b),(c, c),(c, d),(d, c),(d, d)} on the set A={a, b, c, d, e}, which is the following is true? Symmetric а. Reflexive, Symmetric, Transitive Ob. Reflexive, Anti-Symmetric С. Transitive d. Reflexive, Transitive е. O f. Reflexive, Symmetric Anti-Symmetric, Transitive g. O h. Symmetric, Transitive Reflexive, Symmetric, Anti-Symmetric, Transitive i. Oj. Reflexive, Anti-Symmetric, Transitive Reflexive k.Let Sso denote the set of divisors of 80. Let D be the division relation: D={(x, y)|x divides y}. Show that (S0, D) is a lattice and draw its Hasse diagram.Let R be a relation on the set of all integers such that aRb if and only if a + b is even. To show that R is transitive, we must prove that O For any integers a, b and c, if a + b is even and b + c is even, then a + c is even O For any integers a, b and c, if a + b is even and b + c is even, then a +cis odd O For any integers a, b and c, if a + b is even or b + c is even, then a + c is even O For any integers a, b and c, if a + b is even or b + c is even, then a +cis odd To show that R is not transitive, we must find a counterexample satisfies which of the following statement? For some integers a, b and c, a + b is even or b + c is even and a + c is even O For some integers a, b and c, a + b is even or b + c is even and a + c is odd O For some integers a, b and c, a + b is even and b + c is even and a + c is odd O For some integers a, b and c, a + b is even and b + c is even and a + c is even