During the process of the minimization of the following DFA using Equivalence Theorem, O D A 1 1 E B What are the 1-equivalence partition sets? 1 F C 0,1
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- 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}}.Given the following relations: A B C 3 9 8 14 5 8 7 12 r E 0 В 8 S A α В Y 8 Find (i) the intersection of r and s (rs) (ii) the union of r and s (r Us) (iii) the difference between r and s (rs). B 4 5 6 7 C 6 8 10 12Show 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.
- Use the pumping lemma to show that the following set is NOT regular: {ww | w => {a,b}*}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 evenGive 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?
- 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.Consider an equivalence relation R on X={a,b,c}. R= {(a,a), (a,c), (b,b), (c,a), (c,c) } [a]=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.
- Discrete A binary relation R on set A = {1,2,3,4,5,6,7,8} is defined as:(∀x, y∈A) xRy = ((x, y) | x | y) where x | y = x evenly divides y.a. List all members of Rb. Draw a Hasse diagram of the relationc. Explain whether the relationship is LatticeProve the following equalities using equivalence laws: a. ¬(¬rA s) A (r V s) =r b. r→ (-s→ t) = (r A-s)→t Solve this: c. (p →q) →rFor 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.