We proved some facts about arithmetic that you probably feel are "obvious", for example that $0*x = 0$ and $-x = -1*x$. But remember, we didn't prove them about integers or reals or rationals specifically, but for any ring ... so any kind of number system that allows arithmetic. So here's another thing that might seem obvious: $1 + 1 \neq 0$. That's obvious, right????

Part 1: The Boolean Ring

It turns out that the boolean values with $\oplus$ as $+$, and $\wedge$ as $*$ is a ring! Don't take my word though, check for yourself. In other words, verify that the following interpretation is a model (in the first order logic sense) for the ring axioms.

domain: {false,true}
$+$ operator: $a+b$ defined to be $a \oplus b$
$*$ operator: $a*b$ defined to be $a \wedge b$		 Note: Please use T and F for true/false values, not 1/0. "1" means multiplicative identity and "0" means additive identity, and you can't use them until you know which elements have those properties.

axioms of addition axioms of multiplication
  • 1.i: + is associative ... is it? How do you know?
  • 1.ii: + is commutative ... is it? How do you know?
  • 1.iii: there is an additive identity ... what is it? I.e. what is "0" for this ring?
  • 1.iv: each element of the domain has an additive inverse? What's the additive inverse of false? Of true?
  • 2.i: * is associative ... is it? How do you know?
  • 2.ii: * is commutative ... is it? How do you know? ← NOT REQUIRED FOR A RING, BUT IS IT?
  • 2.iii: there is a multiplicative identity ... what is it? I.e. what is "1" for this ring?
  • 2.iv: NOT REQUIRED, BUT ... does each non-zero element have a multiplicative inverse?
  • 3.i: + and * must distribute, i.e. $\forall a,b,c[a*(b+c) = a*b + a*c]$ Is this true for this domain? How can we check?

Part 2

In the boolean ring (Part 1), is it true that $1 + 1 \neq 0$?

Part 3: The Shape Ring

Here is a funny ring (and, yes, I did indeed verify that all the ring axioms hold!). Let's call it the "shape ring". 
domain: {△,○,□,◇}
,    
 + | △  ○  □  ◇ 
---|------------
 △ | □  ◇  △  ○ 
 ○ | ◇  □  ○  △ 
 □ | △  ○  □  ◇ 
 ◇ | ○  △  ◇  □
,    
 * | △  ○  □  ◇ 
---|------------
 △ | △  ○  □  ◇ 
 ○ | ○  □  □  ○ 
 □ | □  □  □  □ 
 ◇ | ◇  ○  □  △
  1. Which element is the additive identity (i.e. what is "0")?
  2. Which element is the multiplicative identity (i.e. what is "1")?
  3. Remember: the "additive inverse axiom" says that for all $x$, ∃y[x + y = 0].

    • The additive inverse of △ is:
    • The additive inverse of ○ is:
    • The additive inverse of □ is:
    • The additive inverse of ◇ is:
  4. All of the theorems we proved for rings last class must hold in the shape ring. For example we proved that for all x, -x = -1*x. Verify that this is true for the shape ring!
  5. You might expect that $1 \neq -1$ should be true for all rings. Is it true for the shape ring? Jusitfy!
  6. Consider the following proposition: $\forall x[ x \neq 0 \Rightarrow x*x \neq 0 ]$
    Is this statement true for all rings? Justify!