algebraic structures

$A=\{0,1,2,3, \ldots\}$ is the set of non-negative integers. Let $F$ be the set of functions from $A$ to itself. For any two functions, $f_1, f_2 \in \mathrm{~F}$ we define $$\left...

Consider the set $$S = \left\{ {1,\,\omega ,\,{\omega ^2}} \right\},$$ where $$\omega $$ and $${{\omega ^2}}$$, are cube roots of unity. If $$ * $$ denotes the multiplication opera...

Which one of the following in NOT necessarily a property of Group?

For the set $$N$$ of natural numbers and a binary operation $$f:N \times N \to N$$, an element $$z \in N$$ is called an identity for $$f$$ if $$f\left( {a,z} \right) = a = f\left(...

Consider the set $$H$$ of all $$3$$ $$X$$ $$3$$ matrices of the type $$$\left[ {\matrix{ a & f & e \cr 0 & b & d \cr 0 & 0 & c \cr } } \right]$$$ Where $$a, b, c, d, e$$ and $$f$$...

Let $$S = \left\{ {0,1,2,3,4,5,6,7} \right\}$$ and $$ \otimes $$ denote multiplication modulo $$8$$, that is, $$x \otimes y = \left( {xy} \right)$$ mod $$8$$ (a) Prove that $$\left...

Let $$\left( {\left\{ {p,\,q} \right\},\, * } \right)$$ be a semi group where $$p * p = q$$. Show that: (a) $$p * q = q * p,$$, and (b) $$q * q = q$$

Let (A, *) be a semigroup. Furthermore, for every a and b in A, if $$a\, \ne \,b$$, then $$a\,*\,b \ne \,\,b\,*\,a$$. (a) Show that for every a in A a * a = a (b) Show that for eve...

Which of the following statements is false?

Let A be the set of all nonsingular matrices over real numbers and let * be the matrix multiplication operator. Then

Some group (G, o) is known to be abelian. Then, which one of the following is true for G?

(a) If G is a group of even order, then show that there exists an element $$a \ne e$$, the identifier $$g$$, such that $${a^2} = e$$ (b) Consider the set of integers $$\left\{ {1,2...