In the last lecture, we saw how you can combine conjunction and disjunction to build up even larger propositions from propositional units. In this lecture, we're going to talk about two properties, commutativity and associativity. Properties that we find in conjunction and in disjunction, but that we don't find in other truth functions like the conditional, which we'll study later this week. So, what are commutativity and associativity? Well, let me start by giving a definition of commutativity. A function of two things is commutative. It has commutativity when it delivers the same result no matter what order it operates on those things. Now, that is a very abstract characterization and that's a characterization that can apply to functions in all different domains. For instance, in arithmetic, there are functions like addition and addition is an example of the commutative function. Addition is a function that can operate on two inputs, you can add two numbers together. And the result that you get when you add two numbers together doesn't depend on the order in which you add the numbers. For example, 2+3 is the same as 3+2. And in general, for any two numbers x and y. X+y is the same as y+x. So, addition is commutative. Addition is an example of a function that has commutativity. Another example of an arithmetical function that's commutative is multiplication. For example, 4 multiplied by 8 is the same as 8 multiplied by 4. More generally, for any two numbers x and y, x multiplied by y is the same as y multiplied by x. So, multiplication is a commutative function. It's a function that has commutativity, but notice not all arithmetical functions are commutative. For instance, subtraction is not commutative. 7-3 is not the same as 3-7. And more generally, for most numbers x and y, x-y is not the same and y-x. That's not true for every assignment of numbers to x and y, but it's true for most. Generally, we can't expect x-y to be equal to y-x. So subtraction is not a commutative function, neither is division. So addition and multiplication are commutative, but subtraction and division are not commutative. Now, the same points can be made about truth-functional connectives in propositional logic. For instance, the truth-functional connective conjunction is commutative. Consider the proposition, I'm standing and waving. That's the conjunction of two propositions, I'm standing and I'm waving. I'm standing and waving is the conjunction of those two propositions. Well, that's the same proposition as I'm waving and standing. They're just two different ways of saying the same thing. Maybe one is a more elegant way of saying it than the other, but they're saying the same thing. What they're saying is the same. In general, the conjunction of any two propositions, p and q is exactly the same as the conjunction of the propositions q and p. Conjunction operates on two propositions and it doesn't matter what order you put the propositions in. P conjunction q is the same proposition, as q conjunction p. So, conjunction is a commutative truth function. The same is true of disjunction. Disjunction is also a commutative truth function. So consider the proposition, I'm standing or waving. That's a disjunction, but it's the same proposition as I'm waving or standing. In general, for any two proposition p, q, the disjunction p or q is the same proposition as the disjunction q or p. So, disjunction is a commutative truth function like conjunction. But later on this week, we'll learn about another truth function that we'll call the conditional. Sometimes it's called the material conditional, but the conditional is not a commutative truth function. With the conditional, the order in which it operates on propositions determines the result that you get. We'll see that later this week when we study the conditional. So, that's commutativity. Now, what about associativity? Well, what's associativity? Here's a definition. A function of three or more things is associative when it delivers the same result no matter what order it operates on those things. So, let's look at some examples of associative functions and of functions that are not associative. First, we'll begin with arithmetic. So just as addition is a commutative function, it's also an associative function. To illustrate, consider a case where addition applies to three things instead of just two. Consider adding 2 to 3 plus 6. Well, the sum of 2 and 3 plus 6, notice that's going to be the same as the sum of 2 plus 3 and 6. Both of them are going to add up to 11. And more generally, for any three numbers x, y and z. The sum of x and y+z is going to be equal to the sum of x+y and z. It doesn't matter whether we group the y and z together and then add that to the x or whether we group the x, and y together and add that to the z. Either way, we get the same result. So, addition is an associative function. It's an example of an arithmetical function that has associativity. The same is true for multiplication. Notice if we multiply 2 by the product of 3 and 6, we get the same number as we get if we multiply the product of 2 and 3 by 6. Either way, we get 36. So in general, for any three numbers x, y and z, the product of x and y times z is the same as the product of x times y and z. So, that explains why multiplication is associative. It's an example of arithmetical function that has associativity, but not arithmetical are associative. In particular, subtraction is not associative. Take any three numbers, x, y, and z. Now, start by subtracting z from y and then subtract the result of that from x. You're going to get a different number in general than if you start by subtracting y from x and then subtract z from that number. So, x-(y-z) is in general, not the same as (x-y)-z. So, subtraction is not associative and neither for that matter is division. So addition and multiplication are associative, just as they're commutative. Subtraction and division are not associative, just as they're not commutative. Analogously with truth functions, we can see that conjunction is associative. For example, consider the proposition I'm standing and he's sitting and waving. Well, that's the same as the proposition I'm standing and he's sitting and he's also waving. In general, for any three propositions p, q and r. The conjunction of p with q and r is going to be the same proposition, as the conjunction p and q with r. Now, just as conjunction is associative so too is disjunction. We can see an example. For instance, if I say, I'm standing or he's sitting or waving. Well, that's the same proposition as I'm standing or he's sitting or he's waving. In general, for any three propositions, p, q and r. The disjunction of p with q or r is going to be the same proposition, as the disjunction of p or q with r. So disjunction, like conjunction is associative. But again, not all truth functions are associative. Later this week, we're going to learn about a truth function called the conditional that is not associative. When the conditional operates on three or more inputs, it matters very much the order in which it operates on those inputs. Because depending on the order, you end up with a different propositional result. So, that's enough about commutativity and associativity. See you next time.