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In addition to the simple operators and, or, and not, we have some complex operators that are used just as frequently. Chief among these are if, material implication, xor, exclusive or, and equals, equivalence or identical. These more complex operators are often used in conjunction with the simple ones to create more elaborate arguments. Material implication, and exclusive or, can both be used to create conditional statements.
This is something we do in natural language. And it's also one of the key components of any computer application. Both these operators allow us to create arguments whose outcomes depend on the conditions of the statements in those arguments. Earlier in the course, I gave you the following example. A waiter asks, do you want fries or salad? And you answer, fries unless everyone is having dessert. In that case, I'll have the salad. If we re-organize this argument using material implication, if.
And negation, not. We get something like this. If, what everyone is eating equals dessert, then, I'll have salad. If, what everyone is eating not equals dessert, then, I'll have fries. This is the thought process that goes through the mind of the waiter, but it's formalized in a way that is more suited for a computer. Humans are able to solve these types of conditional arguments quite easily as long as they are presented with accurate statements. In our earlier example with the question of whether Pooh wants honey or condensed milk with his bread, Rabbit should have used an exclusive or rather than a simple or like this.
Honey exclusive or condensed milk with your bread? In this statement, Pooh has to choose either hone or condensed milk. Both is not an option. The last option, equivalence or identical, is a bit of a strange one. If I present you with ten $1 bills, and one $10 bill, you'll tell me they're equal, right? And you'd be right as long as we were talking about value. But in logic, equivalence is identical.
When you use the equal sign, what you are testing is for an identical result on both sides. A cupcake is only equivalent to the exact same cupcake. A cupcake is not equivalent to any other seemingly identical cupcake or two smaller cupcakes that combined weigh the exact same. In natural language, logical equivalence would be akin to saying something like is exactly. Midnight is exactly at 0000 hours, not at any other time.
The winning lottery numbers are exactly the same as the random draw. The person with my thumb print is identical to me. Because equivalence is so specific, it's often augmented with the use of greater or lesser than to form more common statements. When you're exactly this tall or taller, you can go on the cool rides at the amusement park. When you're exactly 18 years of age or older, you're allowed to vote in elections. In human language and when using computers, we make complex, logical arguments using combinations of all these operators.
It can go something like this: IF I go on a plane, AND fly south for two hours and 45 minutes, AND drive north for 50 minutes, XOR I drive south for 19 hours and 57 minutes, THEN I will end up in Carpinteria. For dinner I can have lasagna, exclusive or pizza, exclusive or milk and cereal.
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