# Boolean operations AND

When the $\times$ sign means AND, the possible results are

1. 0 × 0 = 0
2. 0 × 1 = 0
3. 1 × 0 = 0
4. 1 × 1 = 1

In other words, the result is 1 only if both the left operand AND the right operand are 1.

This is axiom?

• do you mean axiom??? Mar 11, 2017 at 20:00
• We'd rather say that it is the definition of the operator.
– user65203
Mar 11, 2017 at 20:01
• An axiom is a statement accepted without evidence. Mar 11, 2017 at 20:02

What you've written (the truth-table for the AND-connective, essentially) and your summary is correct, but it expresses the definition of AND, but it is not an axiom. The four assignments you show define precisely when $a\times b$ (and in this case $a\land b$) is true, and when it is false.

Similarly, we define the inclusive OR as follows

1. 1+1 = 1

2. 1+0 = 1

3. 0+1 = 1

4. 0+0 = 0

Whenever at least one of the disjuncts is 1, then the disjunction is 1.

The Exclusive Or: a XOR b is defined such that when one and only one of a, b is true, then a XOR B = 1.

This gives us:

1 XOR 1 = 0

1 XOR 0 = 1

0 XOR 1 = 1

0 XOR 0 = 0

• On what this definition is based? Mar 11, 2017 at 20:08
• It's how we define $\land$, or $\lor$. (Just as we define, say $A\subseteq B$ to mean $a \in A \rightarrow a\in B$, and use it consistently.) Mar 11, 2017 at 20:11
• Suppose it is claimed that Jack is your friend and John is your friend. If you are friends with neither, you can judge that claim false. $(0\times 0 = 0)$  If you are friends with Jack, but not John, you can judge the that claim false. $(1 \times 0 = 0)$  If you are not friends with jack, but you are friends with John, you can conclude the claim was false. $(0\times 1 = 0)$  Finally, if you are friends with BOTH Jack and John, the claim is true. $(1\times 1 = 1)$  Mar 11, 2017 at 20:45

A better way to explain what an axiom is:

An axiom is a statement that is treated as true without proof.

These are the assumed premises that you start with to make logical conclusions. What you have there is not an axiom. It's an operation $\times$ which you have defined as the construct AND.

The reason it's not an axiom is that we are restricting an action. In this case we are setting $\times$ as a valid operation within the Boolean algebraic structure we've built. This way if someone tries to use $\times$ they won't come up and say $0\times 0=1$.

By the way you can define $\times$ such that $0\times 0=1$ would be true.