Discrete
1.Β Β Β Β Idempotent laws:
a) π β¨ π β‘ π
b) π β§ π β‘ π
Β
2.Β Β Β Β Associative laws:
a) (π β¨ π) β¨ π β‘ π β¨ (π β¨ π )
b) (π β§ π) β§ π β‘ π β§ (π β§ π )
Β
3.Β Β Β Β Commutative laws:
a) π β¨ π β‘ π β¨ π
b) π β§ π β‘ π β§ π
Β
4.Β Β Β Β Distributive laws:
a) π β¨ (π β§ π ) β‘ (π β¨ π) β§ (π β¨ π )
b) π β§ (π β¨ π ) β‘ (π β§ π) β¨ (π β§ π )
Β
5.Β Β Β Β Identity laws:
a) π β¨ πΉ β‘ π
b) π β§ π β‘ π
Β
6.Β Β Β Β Domination laws:
a) π β§ πΉ β‘ πΉ
b) π β¨ π β‘ π
Β
7.Β Β Β Β Double negation law:
¬¬π β‘ π
Β
8.Β Β Β Β Complement laws:
π β§ Β¬π β‘ πΉΒ Β Β Β Β Β Β Β Β Β Β Β Β Β Β Β Β π β¨ Β¬π β‘ πΒ Β Β Β Β Β Β Β Β Β Β Β Β Β Β Β Β Β¬π β‘ πΉΒ Β Β Β Β Β Β Β Β Β Β Β Β Β Β Β Β Β Β Β Β Β Β Β Β Β Β Β Β Β Β Β Β Β Β Β Β Β Β Β Β Β¬πΉ β‘ π
Β
9.Β Β Β Β De Morganβs laws:
Β¬(π β¨ π) β‘ Β¬π β§ Β¬π
Β¬(π β§ π) β‘ Β¬π β¨ Β¬π
Β
10.Β Absorption laws:
π β¨ (π β§ π) β‘ π
π β§ (π β¨ π) β‘ π
Β
11.Β Conditional identities:
π β π β‘ Β¬π β¨ π
π β π β‘ (π β π) β§ (π β π)
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In this assignment you will write code that will prove both equations for three logical equivalences...
In this assignment you will write code that will prove both equations for three logical equivalences (pick any three except the double negative law). Below is the list of logical equivalences. Please create a program that allows a user to test logical equivalences and have proof of their equivalency for the user. The rubric is below. Submit screen shots of the code, input, and output of the program. Theorem 2.1.1 Logical Equivalences Given any statement variables p, q, and r,...
(b) Use the specified laws and axioms of logic to prove that p ββ q β‘...
(b) Use the specified laws and axioms of logic to prove that p ββ q β‘ (p β¨ q) β (p β§ q). The first step is given. (6 Γ 2 = 12 marks) Step Specified Law or Axiom (i) p ββ q β‘ (~p β¨ q) β§ (~q β¨ p) (ii) (iii) (iv) (v) (vi) (vii) The equivalence law says p ββ q β‘ (p β q) β§ (q β p) and the implication law means p β q...
number a and b 70 Score: B. Bader collin Alhusni_ DATE ΠΡ output E LUBODA RAO+O...
number a and b
70 Score: B. Bader collin Alhusni_ DATE ΠΡ output E LUBODA RAO+O D ot x 1= X X-D=0 1. X+0=X XXX XX'=0 2. x + 1= 1 Idempotent laws: 3. X+X=X Involution law: 4. (X'=X Laws of complementarity: 5. X+X' = 1 Commutative laws: 6. X + Y=Y+X Associative laws: 7. (X+Y) +2=X+ (Y+Z) =X+Y+Z XY YX (XY)Z = X(YZ) = XYZ Distributive laws: 8. XIY + Z) = XY + XZ De Morgan's laws 9....
0/3 POINTS PREVIOUS ANSWERS EPPDISCMATH5 6.4.002. MYN Assume that B is a Boolean algebra with operations...
0/3 POINTS PREVIOUS ANSWERS EPPDISCMATH5 6.4.002. MYN Assume that B is a Boolean algebra with operations + and. The universal bound law for + states that for every a in B, a + 1 = 1. Supply the missing reasons in the following proof for this law. Use only the axioms for a Boolean algebra. Proof: Let a be any element of B. Then: a + 1 = a + (a + a) by the commutative law for + x...
5. Prove each of the following set equalities both by Venn Diagram and by algebraic method....
5. Prove each of the following set equalities both by Venn
Diagram and by algebraic method.
(a) A - (B C) = (A - B)
(A - C)
(b) A - (B C) = (A - B)
(A - C)
(c) A (B - C) = (A
B) - C = (A
B) - (A C)
Hint: To prove the last form, use the equality
A C' =
A (A'
C').
(d) A (B - C) = (A
B) (A...
Let po, P1, ...,Pn be boolean variables. Define ak = (Pk + (ak-1)), where ao =...
Let po, P1, ...,Pn be boolean variables. Define ak = (Pk + (ak-1)), where ao = po. Prove the following boolean-algebra identity using proof by induction and the rules of boolean algebra (given below). Poan = po, for all n > 1. Equivalently this can be written out as: po Β· (Pn + (Pn-1 +...+(p2 + (p1 + po)...)) = po, for all n > 1. (p')=P (a) Commutative p.q=qp p+q = 9+p (b) Associative (p. 9).r=p.(q.r) (p+q) +r=p+(q +...
please solve using all 10 listed bellow: 1. closure property of addition, 2. commutative property, 3....
please solve using all 10 listed bellow:
1. closure property of addition,
2. commutative property,
3. associative property,
4. additive identity property,
5. additive inverse property,
6. closure property of scaler multipication,
7. vector distributive property
8. scaler distributive property,
9. scaler associative property
10. scaler identity property
2. Let V2 = R', the set of all 3-D vectors, with vector addition and scalar multipli- cation defined as follows: β’ if a = (a1, 02, 03) and b = (b.b2,...
I need help with R5 and R8. Thank you! Let R-Z with new addition γ₯ and...
I need help with R5 and R8. Thank you!
Let R-Z with new addition γ₯ and new multiplication O defined as follows. For each a, be R. Addition: ab-a+b-1 Multiplication aOb-a+b-a.b where the operations and are ordinary integer addition, subtraction, and multiplication It can be shown that R is a commutative ring with identity. (a) Verify ring axioms R4, R5, R6, R7, and, BS (First Distributive Law). R5. Existence of Additive Inverses. For each aE R, there exists n e...
Verify the logical equivalences using the theorem below: (p β§ ( ~ ( ~ p β¨...
Verify the logical equivalences using the theorem below:
(p β§ ( ~ ( ~ p β¨ q ) ) ) β¨ (p β§ q) β‘ p
Theorem 2.1.1 Let p, q, and r be statement variables, t a tautology, and c a contradiction. The following logical equivalences are true. 1. Commutativity: p1q=q1p; p V q = 9VP 2. Associativity: ( pq) Ar=p1qAr); (pVq) Vr=pv (Vr) 3. Distributivity: PA(Vr) = (p19) (par); p V (qar) = (pVg) (Vr) 4. Identity: pAt=p:...
In MATLAB: Q1. Solve the following system of linear equations: -4x + 3y +z = -18...
In MATLAB:
Q1. Solve the following system of linear equations: -4x + 3y +z = -18 5x + 12y β 2z = 30 2x - 5y + 6z = 9 Q2. Given the following matrix: 1 2 A= 1-3 5 61 10 -2 3 -9 1 Define A in MATLAB and perform the following operations. Show the result and explain it. a. A*A b. A*A C. A./A d det(A) e. inv(A) Q3. Define A, B, and C in MATLAB and...
In this assignment you will write code that will prove both equations for three logical equivalences (pick any three except the double negative law). Below is the list of logical equivalences. Please create a program that allows a user to test logical equivalences and have proof of their equivalency for the user. The rubric is below. Submit screen shots of the code, input, and output of the program. Theorem 2.1.1 Logical Equivalences Given any statement variables p, q, and r,...
number a and b
70 Score: B. Bader collin Alhusni_ DATE ΠΡ output E LUBODA RAO+O D ot x 1= X X-D=0 1. X+0=X XXX XX'=0 2. x + 1= 1 Idempotent laws: 3. X+X=X Involution law: 4. (X'=X Laws of complementarity: 5. X+X' = 1 Commutative laws: 6. X + Y=Y+X Associative laws: 7. (X+Y) +2=X+ (Y+Z) =X+Y+Z XY YX (XY)Z = X(YZ) = XYZ Distributive laws: 8. XIY + Z) = XY + XZ De Morgan's laws 9....
0/3 POINTS PREVIOUS ANSWERS EPPDISCMATH5 6.4.002. MYN Assume that B is a Boolean algebra with operations + and. The universal bound law for + states that for every a in B, a + 1 = 1. Supply the missing reasons in the following proof for this law. Use only the axioms for a Boolean algebra. Proof: Let a be any element of B. Then: a + 1 = a + (a + a) by the commutative law for + x...
5. Prove each of the following set equalities both by Venn
Diagram and by algebraic method.
(a) A - (B C) = (A - B)
(A - C)
(b) A - (B C) = (A - B)
(A - C)
(c) A (B - C) = (A
B) - C = (A
B) - (A C)
Hint: To prove the last form, use the equality
A C' =
A (A'
C').
(d) A (B - C) = (A
B) (A...
Let po, P1, ...,Pn be boolean variables. Define ak = (Pk + (ak-1)), where ao = po. Prove the following boolean-algebra identity using proof by induction and the rules of boolean algebra (given below). Poan = po, for all n > 1. Equivalently this can be written out as: po Β· (Pn + (Pn-1 +...+(p2 + (p1 + po)...)) = po, for all n > 1. (p')=P (a) Commutative p.q=qp p+q = 9+p (b) Associative (p. 9).r=p.(q.r) (p+q) +r=p+(q +...
please solve using all 10 listed bellow:
1. closure property of addition,
2. commutative property,
3. associative property,
4. additive identity property,
5. additive inverse property,
6. closure property of scaler multipication,
7. vector distributive property
8. scaler distributive property,
9. scaler associative property
10. scaler identity property
2. Let V2 = R', the set of all 3-D vectors, with vector addition and scalar multipli- cation defined as follows: β’ if a = (a1, 02, 03) and b = (b.b2,...
I need help with R5 and R8. Thank you!
Let R-Z with new addition γ₯ and new multiplication O defined as follows. For each a, be R. Addition: ab-a+b-1 Multiplication aOb-a+b-a.b where the operations and are ordinary integer addition, subtraction, and multiplication It can be shown that R is a commutative ring with identity. (a) Verify ring axioms R4, R5, R6, R7, and, BS (First Distributive Law). R5. Existence of Additive Inverses. For each aE R, there exists n e...
Verify the logical equivalences using the theorem below:
(p β§ ( ~ ( ~ p β¨ q ) ) ) β¨ (p β§ q) β‘ p
Theorem 2.1.1 Let p, q, and r be statement variables, t a tautology, and c a contradiction. The following logical equivalences are true. 1. Commutativity: p1q=q1p; p V q = 9VP 2. Associativity: ( pq) Ar=p1qAr); (pVq) Vr=pv (Vr) 3. Distributivity: PA(Vr) = (p19) (par); p V (qar) = (pVg) (Vr) 4. Identity: pAt=p:...
In MATLAB:
Q1. Solve the following system of linear equations: -4x + 3y +z = -18 5x + 12y β 2z = 30 2x - 5y + 6z = 9 Q2. Given the following matrix: 1 2 A= 1-3 5 61 10 -2 3 -9 1 Define A in MATLAB and perform the following operations. Show the result and explain it. a. A*A b. A*A C. A./A d det(A) e. inv(A) Q3. Define A, B, and C in MATLAB and...
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