# Logic and Proof

# Conjunctions

There are also ways to connect two statements to make a bigger statement, like Legos. One is by using the word "*and*," and the result is called the **conjunction** of the two statements. Sort of like the kind in School House Rock, but not exactly. The statement "*A* and *B*" is true exactly when both *A* is true and *B* is true. From a more pessimistic point of view, "*A* and *B*" is false as soon as one of *A* or *B* is false.

### Sample Problem

Sam is 6 feet tall and has brown hair. He also has blue eyes, his favorite character from *The Lord of the Rings* is Samwise Gamgee, his favorite album is *Abbey Road*, he prefers red apples to green, he collects pewter figurines with little crystals embedded as dramatic accents, and he wakes up in the morning screaming "Wednesday!" every Wednesday. Is the statement, "Sam is shorter than 6'3" and has brown hair" true?

Since the statement is a conjunction, we check each atom individually. Does Sam have brown hair? Yes, it says so right in the first line up there. Is Sam shorter than 6'3"? Yes, his height is an even 6 feet. Since both parts are true, we conclude that the whole statement is true.

### Sample Problem

Is the statement "Sam's favorite character from *The Lord of the Rings* is not Gandalf the Grey and his favorite album is not *Abbey Road*" true?

Again, let's take a look at each atom. The first atom is a negation, saying that it's not true that Sam's favorite character is Gandalf. That's correct, because his favorite LOTR character is Samwise. The next atom is also a negation that says that it's not true that his favorite album is Abbey Road. But Sam's favorite album *is* *Abbey Road*. So the negation is wrong. Since the statement is a conjunction, the whole thing is wrong.