Let $X = \{x, y\}$ be a set with associative and commutative operations $+$, $\cdot$ defined as follows \begin{align} x+x&=x, & x+y&=y, & y+y&=x \\ x\cdot x&=x, & x\cdot y&=x, & y\cdot y&=y. \end{align} Verify that X with the operations is a field
So I know I have to verify additive identity and inverse, multiplicative identity and inverse, and distributivity,not sure how to check this here though?
$x$ and $y$ are the only elements in the set. We see that $x+x=x$ and $x+y=y$, so (together with commutativity of $+$) we find $x$ as the additive identity. Similarly, we get $y$ as the multiplicative identity.
Checking inverses, we find that $-x=x$, $-y=y$ and $y^{-1}=y$ ($x$ excluded from the multiplicative inverse check, being the additive identity). Finally, it should be easy to check via casework (substitute $x,y$ into $a\cdot(b+c)=a\cdot b+a\cdot c$) that distributivity is satisfied. Hence the given structure is a field.