Let * be the operation on the cartersian product of real numbers R *R such that (a,x) * (b,y) = (ab,x+y). Then the inverse of (a,x) is O A. -a,

Q: Prove that the function g: Real #s to Real #s* defined by g(x)=2x is an injective homomorphism that…

A:

Q: Q3 Iet X-{mingP;q } and ΦΧ 3. min is 2 asigma algebra over over X ?

A: Given that  X = {m, n, p, q} and ∑=ϕ, X, {m}, {n}, {p, q}, {m, n}, {n, p, q}, {m, p, q} we have to…

Q: Consider the following relation < over reals: x < y iff (x– y) e Z. Prove that it is an equivalence…

A:

Q: Define A: C → C° by A(z,, Z2, Z3) = (a,z1, azz2, lzz), «; e C. When is A a positive operator ?

A: A is a positive operator if all the eigen values are non negative Therefore, T(1, 0, 0) = (α1, 0,…

Q: Prove by structural induction that F C F2

A: Given that, F1 such that, 3∈F1, n∈F1⇒n2∈F1. and F2 such that, 3∈F2, n,m∈F2⇒nm∈F2. Consider Pn be…

Q: A relation < is defined on R? by (x1, x2) < (y1, y2) if and only if x1 < yi and x1 + x2 2 yı + y2.…

A:

Q: A relation R is defined on the set of real numbers by xRy if and only if x-y is a multiple of 3.…

A:

Q: PROVE THE FOLOWING: PROVE thtAT IF AND ONLY %3D IF is REAL

A:

Q: Give an example of two equivalence relations R and S on

A: Consider that R and S are equivalence relations on A=1, 2, 3.

Q: Let x={1,2,3,4} and y={1,2,3,4,5,6,7,8,9,10}   and "f" from X to Y is given by…

A:

Q: Derive that (x + y)(x' + z)(y + z) = (x + y)(x' + z) by using boolean algebra.

A: Kindly go through the solution and let me know in case of any doubt or further clarification in the…

Q: Let A be an n × n symmetric negative definitematrix. Show that the leading principal submatrices of…

A:

Q: Let . be the operation on the cartersian product of real numbers R • R such that (a,x) * (b.y) =…

A: Given (a,x)*(b,y)=(ab,x+y)

Q: Let G={1,-1,i,-i} with usual multiplication then G is isomorphic to Select one: a. Z, x Z, b. (Z, +)…

A:

Q: Let f: A → B, g: B →C be bijections. Then prove that (gof) =f'og

A:

Q: 2) Let R real numbers, define = x,Y1 +4x2y2 + 6x3Y3 for x = (x1,x2, X3) and y (y1,y2, Ya). Is inner…

A: Let V be a vector space over ℝ.An inner product &lt;·,·&gt; is a function V×V→ℝ with the following…

Q: (c) Show that T-l is not bounded, and explain why this is not in contradiction with the Open Mapping…

A: Af first we will show that T is bijective and hence its inverse T -1 exists. Then we will find that…

Q: Let f, g,h: A→ A be any bijective functions. Then (f ogoh)-

A:

Q: Prove or disprove that ⌈x + y⌉ = ⌈x⌉ + ⌈y⌉ for all real numbers x and y.

A: According to the given information, It is required to prove or disprove that:

Q: Let V = {(x,y)x,y € R}. Define addition on V as follows: (x,y)+(x)=(x +2x, 1,y,+Y,+2) Is addition…

A: Given:  Let V=x,y|x,y∈ℝ. Also, the addition on V is defines as x1,y1+x2,y2=x1+2x2-1,y1+y2+2.

Q: Prove that R is not reflexive.

A:

Q: Prove that f(x) is reducible over Z3. Write f(x) as a product of irreducible factors.

A:

Q: + Z,[r] be an evaluation homomorphism. Use Fermat's Theorem to evaluate Let o, : Z, O3(x231 + 3.r17…

A:

Q: Consider the binary relation R defined on (N* x N*) by: V (X1. Yı). (x2. Y2) E N* x N*, (x1. Yı) R…

A: Order relation (definition)-: A relation R is called an order relation if it is reflexive,…

Q: Let f be the function from {a, b, c, d} to {1, 2, 3, 4} with f(a) = 4, f(b) = 2, f(c) = 1, and f(d)…

A: For bijection, the function needs to be one to one and onto.

Q: hat subring R[x², x°] not isomorphic to R[x].

A:

Q: a) Show that this definition of the inner product satisfies both of the above conditions.

A:

Q: Show that the function f: Real #s to Real #s defined by f(x)=x2 is not a homomorphism.

A:

Q: Let R be an integral domain with unity. State what it means for “being associates” to be transitive,…

A:

Q: Show that  ℝ2 = span([1 1], [1 -1])

A:

Q: Let f,g,h: A→ A be any bijective functions. Then (fogoh)-1 f-1og1oh-1 h-1og-1of-1 O is equal to the…

A:

Q: Let f, g,h: A A be any bijective functions. Then (f ogoh)-1 h-1 og-1of-1 Ois equal to the above None…

A: Given that                     f,g,h:A→A be any bijective functions. Then we have to find the…

Q: 3. Construct a bijection f: [0, 1] → (0, 1) UN.

A:

Q: prove R is not homomorphic to R^2

A:

Q: Examine the continuity of if 0 a at

A: We need to examine the continuity of the given function f(x) at x=a.

Q: If X is inner product space over real numbers then the inner product 00 200

A: Definition of inner product space: A vector space is called inner product space when it is equipped…

Q: Show that the relation R in the set {1, 2, 3} given by R = {(1, 2), (2, 1)} issymmetric but neither…

A: Given, relation R in the set {1, 2, 3} given by R = {(1, 2), (2, 1)}.

Q: Prove or disprove that [x + y] = [x] + [y] for all real numbers x and y.

A: we have to prove or disprove that [x+y]=[x]+[y] for all real numbers x and y.

Q: Let X = Rand def ine a function on R x R to R as = eTY verify all conditions of inner product if it…

A:

Q: Let * be the operation on the cartersian product of real numbers R* R such that Then the (а,х)*…

A:

Q: Let a, b, c, d be real numbers where c and d are not both zero. Let X = {x E R | cx + d + 0}. The…

A: "

Q: A complwx number z is a number of the form z=x+iy, x,yER

A: Both the statements are TRUE

Q: 3. Use the cross product properties to prove the relation (a - b) × (â+b) = 2(â × b) given a = (1,…

A: Given,a=i-3j+2kb=-2i+0j+k

Q: Prove that the product xy of two real numbers x and y is nonnegative if if and only if the absolute…

A:

Q: , Let Z be the set of integers. Consider R = {(x, y) e Z x Z: 7 divides x-y} Z x Z. Show that R is…

A:

Q: Prove that continuity is preserved under addition and scalar multiplication

A:

Q: prove or disprove : If F:(R,+) (R\{0},.), defined by f(x)= x +2 is an isomorphism.

A:

Q: Find a bijection from the set of positive integers Z+ to the set of integers Z (Thus, conclude Z+ =…

A: the set of positive integers Z+ to the set of integers Z . the function is defined..