Question III: Let M₁ and M₂ be two R-modules and let r E R. Let M₁ X M₂ = {(x,y) |x E M₁, y € M₂} with operations (x₁, y₁) + (x₂,Y₂) = (x₁+x₂, Y₁ + y₂) and r. (x, y) = (r.x, r.y) Show that M₁ X M₂ is an R-module.

Q: To solve the algebraic equation y – y? + € = 0, we set /2, | y = e2y1 + eY2 + e/2y3+... , then to…

A: Given the algebraic equation, y3-y2+∈=0 we set, y=∈12y1+∈y2+∈32y3+....... y=∈12y2=∈y12y3=∈32y13

Q: Nice property Prove that 1logbc21logcb2= 1, for b 7 0, c 7 0, b ≠ 1, and c ≠ 1.

A: 1logbc2·1logcb2=1 for b > 0, c > 0, b ≠ 1, and c ≠ 1.

Q: Using the formal axiomatic system prove the following theorem (A→ B) →(--B →-A)

A: This question is about application of axiomatic system

Q: Show that following formulas can be deducted in Hilbert System (It is allowed to use Deduction…

A: We have to prove the given formulas by Hilbert system

Q: Q1 Logical Equivalence Prove the following equivalence: (pq)^(-V p) = p^r Q1.1 Truth tables Use a…

A:

Q: Vx, A(x) → B(x) 3x, A(x) ^ C(x) Vx, C(x) → D(x) ..3x, B(x) A D(x)

A:

Q: For circuit M, find out what orders its nonzero ones have elements and find their divisors of zero,…

A:

Q: how to find the value of U(n) on abstract algebra example U(8) = {1,3,5,7}

A: By the definition , U(n)={ a: a<n and gcd(a,n)=1}

Q: Prove that N× N × N× N is countable.

A:

Q: True

A: False.

Q: The product of 2r³ + 3x² +1 and 2x² + 4 in z3 [x]/ is O 5x^2+ 5x +1 O None O x^2+2 4x+1

A:

Q: We define an addition operation on V by (a + bv2) + (c+ dv2) := (a + c) + (b + d)/2, for a +bv/2, c+…

A: The given set is,V=a+b2:a,b∈ℚThe addition and scalar multiplication on V is as…

Q: Suppose V is a complex inner product space. Prove that every normal operator on V has a square root.…

A: To prove: Every normal operator on V has a square root.

Q: bounded Decide unbounded the doma n

A: (As per bartleby guidelines, for questions containing multiple subparts, only first three subparts…

Q: Find the order of (3, 4) in Z4 × Z6

A: Z4={ 0,1,2,3 } mod 4Z6={ 0,1,2,3,4,5 } mod 6Order (a,b) in Zm*Zn = LCM { order( a), order (b)}

Q: n n-r+1 (a) Prove that "C, =""C,-1 (b) Prove that "C, r

A:

Q: Dimensiones of a box

A:

Q: The product of 2x³ + 3x² + 1 and 2x² + 4 in z5[x]/ is: 5x^2+ 5x +1 O None X^2+2 4x+1

A:

Q: Prove that A CB P(A) C P(B).

A: We have to prove that

Q: urses / MATH2100N / WORKSHEETS (CALCULUS - II) / WORK SHEET # 1 (CHAPTERS- Remainder If x2 - 12x 13…

A: Explanation: Given that: x2x2-12x-13     Calculation: On dividing x2 by x2-12x-13 we get the…

Q: Let =[2,0}, then f is o- algebra.

A: Given a statement

Q: The translation of ABCD to A'B'C'D' is given by (x+[?1.y-[ ]). A D. -7 6 -5 4 5. A ID' Enter 3)

A: To find:- Need to find translation  figure  coordinate rule from given figure

Q: QUANTIFICATION True or False (FOR ALL x)(FOR ALL y), x, y are any integers: x + y = 0.

A: True or False (FOR ALL x)(FOR ALL y), x, y are any integers: x + y = 0.

Q: Using an example, demonstrate how the conjunction rule of inference can be used to  prove a theorem

A: Conjunction rule of inference states that,        p       q_______∴ p∧q If the statement p is true…

Q: (r+1)2" = 3"-(2n +3) %3D r30

A: We will answer the first question since we only answer one question at a time. Please resubmit the…

Q: Show that for any nez, n is erther odd or even

A: Suppose P(n) is a given statement, then P(n) is true for all positive integer if: P(1) is true. If…

Q: Proof by induction that 0+0=0+0   where 0 represent x. x+0=0+x

A:

Q: 1337 i+1` Compute ()

A:

Q: Show that if a ú an odd integer, thn a =1 (md 16).

A:

Q: Proof Let

A: Since you have posted a question with multiple sub-parts, we will solve first three subparts for…

Q: Let 9={a: aER}, then 9 is Borel algebra.

A: False

Q: Which of the following statements is/are always true? * Method of Variation of Parameters is…

A: According to the question, the statements:Method of Variation of Parameters is applicable if Method…

Q: Rodrigues formula prove

A:

Q: Which Theorem you can use to determine the unknown coefficient so that the given linear expression…

A:

Q: Given B = v1 {(0, 1, 1, 1)}, v2 = {2,1, –1, –1}, v3 {(1,4, –1,2)}, v4{(6,9, 4, 2)} B' = wi {(0, 8,…

A: In this problem we used Gram-Schmidt orthogonalization  process.