b) Prove that if x

Q: Let F be a field. Prove or disprove: For a, b, c, d E F we have (a+b)+2c-d=2c + (b-d) + a.

A:

Q: Give an example of an extension of fields E/F with [E : F] = 4 and 7 € E, 7 & F such that the set {a…

A: Consider  the  field  F=  and  E=Then, [F:E]=[]                = [Using  Tower  theorem ]…

Q: Let A = {1, 2, 3, 4} and let f:A ->A be defined as {(1, 3), (2, 4), (3, 1), (4, 3)}. Is f…

A:

Q: If F is a field of order n, what is the order of F*?

A: Let F be a finite field  then its order is pn

Q: Prove that F2 is a field.

A: note : Here we will take the standard definition of set F2 = Z/2Z which is integer modulo 2   As per…

Q: Let X = {o, p, s, w}, Y = {a,b, e, g, k}, and Z = {1, 2, 3, 5, 7, 9}. Let ƒ : X → Y where f = {(o,…

A: X=o,p,s,wY=a,b,e,g,kZ=1,2,3,5,7,9f:X→Yf=o,a,p,g,s,e,w,bg:Y→Zg=g,3,e,2,b,9,k,7,a,1

Q: Is Q [x]/⟨x2 -5x + 6⟩ a field? Why?

A:

Q: 2. Let V be the set R regarded as a vector space over the field Q with the usual operations. Show…

A: To show that the set ππk,k=0,1,2,... is linearly independent in the vector space V over the field Q…

Q: (6) Let F be a field and x € F. Prove that -(-x) = = X.

A: Solution: Given that

Q: Let D= {1, 2, 3, 4). Give an example of a relation W on D such that W is anti- symmetric but not…

A:

Q: 7. Show that the function f : R → R* defined by f(x) is not onto. x2 +1

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Q: Exactly one of the following pairs consists of two vector spaces over the field R that are not…

A: Step 1: Step 2: Step 3: Step 4:

Q: If f:{1,2,3}->{a,b}, f(1)=a f(2)=b f(3)=a then explain why f is onto but not one to one

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Q: Assume F1, F2, · · ·, Is U₁ Fi a field? i=1 2 Fn ... is an infinite sequence of fields with F₁ C F₂2…

A: