CS 448 HW 4

Question: 1 Let p and q be propositions then apply the laws of logic to determine whether or not the following statement is a tautology: ~(p → ng) – np = (q Ap). Question: 2 If A = {2,4} and B = follows: {1,4, 5}. Define a binary relation R1 and R2 from A to B as R1 = {(a, b) e (A x B)| a > b}, R2 = {(a, b) e (A x B)| a = b}. a 3D Then: ) Find the ordered pairs in R1 and also find the domain as well as range of R1. (ii) Find the ordered pairs in R2 and also find the domain as well as range of R2.

Given the following CFG a) Convert the CFG to an equivalent CFG in GNF. b) Construct an NPDA that accepts L(G).

Let A = {a, b, c, d} and R = {(a, a), (b, b), (c, c), (d, d), (a, d), (d, a), (b, c), (c, b)} an equivalence relation on A. Select the following sets that are equivalence classes of R {a, b} {a} {a,b,c} O{c, b} O{a, d} {a,b,c,d} {d} {c, d}

2-Rewrite the following BNF to give + precedence over * and force + to be right associative → = →a|b|c + | → * | → ( ) |

Given the following CFG a) Convert the CFG to an equivalent CFG in GNF. b) Construct an NPDA that accepts L(G).

2. 1 9 ef Consider the relation R on Z² given by: R= {((a,b), (c, d)) : a +d = c+b} Prove that R is an equivalence relation, and determine 3 elements of [(1,3)], the equivalence class of (1, 3).

p.609, icon at Example 2 #1. (a) Verify that the following is an equivalence relation on the set of real numbers: aRb → [a] = [b], where [x] is the floor of x. (b) Describe the equivalence classes arising from the equivalence relation in part (a).

3 CNF Step 3Perform step three of converting the following CFG’s into CNF by removingunit rules.3.bV = {S,A,B,C,D}, Σ = {a,b,c}, S = S, R =S →A | (the square is epsilon)A →BCB →BD |bbC →CD |ccD →B |C

Please explain very clear Explanation got thumbs-up

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.

2. Consider the following language: {r € {a,b}* | #a(x) < 2#b(x)} (a) Give a CFG for this set. (b) Prove that the given CFG is correct.

c) Let R be the following equivalence relation on the set A = {1, 2, 3, 4, 5, 6}: R = {(1, 1), (1, 5), (2, 2), (2, 3), (2, 6), (3, 2), (3, 3), (3, 6), (4, 4), (5, 1), (5, 5), (6, 2), (6, 3), (6, 6)}. Determine equivalence classes of R?

3 CNF Step 3Perform step three of converting the following CFG’s into CNF by removingunit rules.3.aV = {S,A,B}, Σ = {a,b}, S = S, R =S →AA →AA |AB |A |B |aBB →BB |Bb |b

Are the functions one to one or onto only? I. f : N → Z : x ↔ x II. f : N → N : x ↔ x III. f : Z → N ∪ {0} : x ↔ |x|

Is each of the following statements true or false? Please explain.  a. If a signature σ is infinite (i.e. contains infinitely many symbols), then a σ-formula can have infinite length.  b. Let σ := (c, f, ·, <) be a signature, where c is a contant symbol, f is a unary operation symbol, · is a binary operation symbol, and < is a binary relation symbol. Then ϕ(y, z) := ∀x(f(x) · c < x) is an extended σ-formula whose interpretation in each σ-structure is a binary relation.  c. For any signature σ and any σ-structure A := (A, σA), if a set D ⊆ Ak is definable (A-definable here), then it is P-definable for some finite P ⊆ A.

FEE*, define o =T if for all w E E*, ow E L(M) if anc M). Show that = is an equivalence relation on D".

Determine with explanations and examples (where appropriate) if each of the following is a trivial functional dependency, where O is the empty set, and A # 0, (а) А > Ф (b) Ф->А (с) Ф>Ф

Question 1 Use the pumping lemma to prove that the following languages are not regular: {uvu : u ∈ {a, b} ∗ , u does not equal (the equal sign with a line through it symbol) ǫ, v ∈ {a, b} ∗}. Full explain this question and text typing work only

Ans 1 d

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.

Let G = ({S,A,B,C,D,E,Z},(0,1),R,S),where R = {S → E | Z; E → A | C; A → 01B | 0A | ; B → 1B |10A; C →10D |1C |; D →01C |0D; Z →0Z1 |}.(a) Describe the language L (in English) that is generated by the CFGG.(b) Prove that the language L generated by the CFG G is not regular.Hint: use the fact that regular languages are closed under unionand prove that one component of the language is not regular bythe Pumping Lemma.

2. Eliminate all e-productions in the following cfg G: G= ({S,A,B}, {a,b,c}, {S→Aa|BBBA, A→BS|ɛ, B-cScɛ}, S).

Let L be L(A), where DFA A = (Q,T, 8, q0, F) over alphabet T = {a, b}. Let h be the homomorphism defined by h(0) = aba and h(1) = €. For the DFA A shown in the figure below, define a DFA B = (Q,E, Y, q0, F) over alphabet E = {0,1}, where h-1(L) = B. Toearn a full credit, the definition for B must be both complete and correct. To solve this problem, you must answer all components for the DFA B. a В A b a h(0) = aba h(1) = €

1. Suppose V = {S, A, a, b}, T = {a, b), S is the start symbol with productions S→ bS, s → gA A → aS, A → RA, A →g, S -b. Find a derivation of each of the following. a) bbabbab b) bbbaab

Discrete Structure

PLEASE TYPE ONLY IF NOT YPE THEN DO NOT DO THIS*** ONLY DO (B VIA D) FOR EXERCISE 8.1.3***

Minimization of a logical function with Weich maps:  Vm(0,1,2,3,4,5,6,7,10,13) Fill in the map by setting the canonical form of the function; Group the members (units) according to the described rules;

answer the following boxes

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Computer science

. Determine whether the CFGs given are ambiguous or not                                                                                                                                                           S→ a|aSb|Y      Y→ Yb|a|aY note: subject: compailer concept deptt: cs/It

CS 1105 Chapter 3 Question 7.4 Construct an NFA model that represents the language by the regular expression: x(y ∪ (xy)∗)(x ∪ yy)∗xy

3. Construct npda’s that accept the following regular languages:(a) L 1 = L (aaa ∗ bab).(b) L 2 = L (aab ∗ aba ∗ ).(c) The union of L 1 and L 2 .(d) L 1 − L 2 .(e) The intersection of L 1 and L 2 .

DISCRETE STRUCTURE