Abstract Algebra

Show that a group G is abelian if for all elements a and b in G, a * b = b * a.

Determine whether the following mappings represent a function: Inputs: \{2, 4, 5\}, Outputs: \{-1, 0, 4\}.

Determine whether the following mappings represent a function: Inputs: {-4, 0, 8}, Outputs: {1, 2, 5, 7}.

Determine whether the following mappings represent a function: Inputs: {-3, -2, 0, 4}, Outputs: {-7, -5, -3}.

Describe an injection from a set $X = \{x_1, x_2, \ldots, x_n\}$ with $n$ elements to the set $\mathbb{Z}$ of integers, where $\theta(x_i) = i$.

Let $\theta: \mathbb{R} \to \mathbb{N}_0$ be a mapping given by $\theta(x) = |x|$. Is $\theta$ injective? Explain.

List the elements of the complete inverse image $\theta^{-1}(5)$ under the mapping $\theta(x) = |x|$.

Find the inverse of 2 modulo 9.

Determine if 3 has an inverse modulo 9.

Find the inverse of 4 modulo 9.

Determine if 6 has an inverse modulo 9.

Find the inverse of 8 modulo 9.

Find the kernel of the homomorphism $\phi: \mathbb{R}^* \to \mathbb{R}^*$ defined by $\phi(x) = x^2$.

Compute $(45 + 38) \mod 7$ by modding first.

Compute $(45 \times 38) \mod 7$ by modding first.

Given two polynomials in $Z_2[X]$, add them and simplify the result.

Add $3X^3 + 2X^2 + 1$ and $3X^3 + X^2 + 2X + 2$ in $Z_4[X]$ and simplify the result modulo 4.

Prove that the function $f(x) = 3x - 2$ is injective.

Prove that the function $f(x) = x^2$ is not injective.

Prove that the function $f(x) = 5x + 2$ is surjective from the reals to the reals.