Simple lattice proofs Suppose that (X,≤) is a lattice and x,y ∈ X. Prove that: 
x ∧ ( x ∨ y ) = x
I'm having trouble understanding how these types of lattice proofs should be structured. I've looked at similar ones and found this approach: 
Let a = x ∧ ( x ∨ y )
Then a ≤ x and a ≤ ( x ∨ y )
But, I'm not sure what "a" is exactly? I have a whole set of these types of proofs to do but I don't have an example to go off of, so if someone could walk me through this one, it would be super helpful. 
 A: $a$ is just an abbreviation for $x\land(x\lor y)$ that they introduced so they don't have to keep writing that long expression.
The fact that $a\le x$ and $a\le (x\lor y)$ follows from the defintion of $\land$ as meaning the greatest lower bound (if it is a lower bound, it is less than both of them).
That $a\ge x$ follows from the fact that $x$ is less than or equal to both $x$ and $x\lor y,$ and thus it is a lower bound for them. And thus $a\ge x$ since $a$ is the greatest lower bound.
A: Your first strategy could be to unwrap definitions, and for this problem no more is needed.

Join:
  $a\vee b$ is an element satisfying $a\vee b\leq a$, $a\vee b\leq b$, and if $z$ satisfies $z\leq a$ and $z\leq b$, then $z\leq a\vee b$

$\phantom{}$

Meet: 
  $a\wedge b$ is an element satisfying $a\wedge b\geq a$, $a\wedge b\geq b$, and if $z$ satisfies $z\geq a$ and $z\geq b$, then $z\geq a\wedge b$

$\phantom{}$
$x\vee y$ is an element such that $x\vee y\leq x$, $x\vee y\leq y$ and if $z\leq x$ and $z\leq y$, then $z\leq x\vee y$.
Now, just verity that $x$ satisfies the definition of $x\wedge (x\vee y)$.
You have that $x\geq x$, $x\geq (x\vee y)$, by definition of $x\vee y$ above,
If $z\geq x$ and $z\geq (x\vee y)$, then in particular $z\geq x$.
