"Or" vs. "Xor" in Math

OR and XOR are two different operations, although they are similar in some ways. Here's how they work.

OR evaluates to True if one or more of the inputs are True, and False if they're all False.

\begin{matrix} & & A\
& & False & True\
B & False & 0 & 1\
& True & 1 & 1 \end{matrix}

XOR is the same as OR, but it evaluates to False if both inputs are True.

\begin{matrix} & & A\
& & False & True\
B & False & 0 & 1\
& True & 1 & 0 \end{matrix}

That's the difference between "OR" and "XOR"! :)

It may be useful to know for the sample problems below that $\lor$ is a logical OR and $\land$ is a logical AND.

OR vs. XOR

When describing binary operations, OR and XOR (exclusive or) are different. Intuitively OR means that one is true and the other isn't, but as an operator, it's slightly different.

Logical Operator

OR takes two values, $p$ and $q$, and if at least one is true, then it returns true. XOR takes those two values again, but if exactly one is true (and the other is false), it returns true.

Here is the truth table for OR and XOR:

$\begin{array}{c|c|c|c} p & q & p \,\,\, \text{OR} \,\,\, q & p \,\,\, AND \,\,\, q \\ \hline 0 & 0 & 0 & 0 \\ 0 & 1 & 1 & 1 \\ 1 & 0 & 1 & 1 \\ 1 & 1 & 1 & 0 \\ \end{array}$

We can use the symbol $\lor$ to describe OR and $\oplus$ to describe XOR.

As a set operator

You can apply OR and XOR to sets too! Let us take two example sets $A$ and $B$. OR is just the union of the sets, denoted by $A \cup B$, which is the set of all distinct elements in the set.

Analogous to its logical counterpart, XOR is also known as the symmetric difference of the two sets, and is the elements that are in one set but not both. It can be denoted by $A\oplus B$ and is equivalent to $(A \cup B) \backslash (A \cap B)$ (think about why this is true!).

Let's give a few examples: $$\{1,2,3\} \cup \{2,4,5,10\} = \{1,2,3,4,5,10\}$$ $$\{\text{cat, dog, pig}\} \oplus \{\text{dog, cat, penguin}\} = \{\text{pig, penguin}\}$$

Now, try these on your own to make sure you understand!

That's it. I hope you enjoy your new knowledge, and thanks for reading!

In common English, "or" can sometimes have an ambiguous meaning. However, in mathematics, that will not do, so we create two separate operations to cover everything that "or" could possibly mean.

The first of these is the inclusive or, denoted $\lor$. That means that either or both statements connected by the $\lor$ can be true.

$A \lor B$ is true $\Leftrightarrow$ $A$ and/or $B$ is/are true.

$A \lor B$ is false $\Leftrightarrow$ $A$ and $B$ are both false.

On the other hand, the exclusive or, denoted, $\oplus$, means that exactly one of the two statements involved must be true.

$A \oplus B$ is true $\Leftrightarrow$ $A$ is true and $B$ is false or $A$ is false and $B$ is true.

$A \oplus B$ is false $\Leftrightarrow$ both statements are true or both statements are false.

In terms of set theory, $\lor$ is the logical equivalent of the union of two sets, and $\oplus$ is the logical equivalent of the symmetric difference of two sets. Therefore, for an arbitrary $x$:

$[(x \in A) \lor (x \in B)] \Leftrightarrow (x \in A \cup B)$

$[(x \in A) \oplus (x \in B)] \Leftrightarrow (x \in A \Delta B)$

Q & A on Or vs. Xor

Q: How are these operators represented on Venn Diagrams? A: A Venn Diagram has three regions: two for each of the two "categories" separately and a third for the overlap. "Or" represents the entire diagram; "Xor" represents the entire diagram minus the middle section.

Q: How are these operators represented in set theory? A: Let the sets in question be $A$ and $B$. "Or" can be represented in one of these ways, where $x$ is an element in relation to sets $A$ and $B$:

$x \in A \cup B$

$(x \in A) \lor (x \in B)$

In a similar vein, the "Xor" operator can be written as

$((x \in A) \lor (x \in B)) \land (x \notin A \cap B)$

$x \in (A \cup B) - (A \cap B)$

Q: Why would anybody want to write these operators out in terms of set theory?

A: It makes writing set containment proofs much easier. For example, here is a proof that the two expressions above are logically equivalent:

$x \in (A \cup B) - (A \cap B) \Leftrightarrow$

$(x \in A - A \cap B) \lor (x \in B - A \cap B) \Leftrightarrow$

$((x \in A) \land (x \notin A \cap B)) \lor ((x \in B) \land (x \notin A \cap B)) \Leftrightarrow$

$(x \notin A \cap B) \land ((x \in A) \lor (x \in B))$

See, isn't that easier than writing things out in words for every step?

Write the expression below using or and xor notation.

In addition to $x$ being equal to $4$, we know that $y$ is equal to either $2$ or $-2$.

Write out the following statement using words.

$((x = 7) \land (y = 3)) \lor ((x = 5) \land (y \neq 3))$