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Variations Using Statements

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Negations

Every statement has a negation. Usually the negation of a statement is
simply the same statement with the word "not" before the verb. The negation of
the statement "The ball rolls" is "The ball does not roll." By definition, the
negation of a statement has the opposite truth value of the original
statement. The negation of *a* statement *a* is âàü*a* (read "not *a*").

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Conjunctions

When two statements are combined with the word "and" the combination of those
statements is called the conjunction of two statements. For example, the
conjunction of the two statements "The weather is rainy" and "the ground is wet"
is the single statement, "The weather is rainy and the ground is wet." The
conjunction of two statements *f* and *g* is symbolized like this:

Figure %: The symbol for the conjuction of two statements

The truth value of a conjunction, of course, depends on the truth values of the
statements that were joined to form the conjunction. A conjunction is only true
if both of the original statements are true. Otherwise, the conjunction is
false.

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Disjunctions

When two statements are joined by the word "or", their combination is called a
disjunction. The disjunction of the two statements in the previous
paragraph is "The weather is rainy or the ground is wet." The symbol for the
disjunction of statements *f* and *g* looks like this:

Figure %: The symbol for the disjunction of two statements

The disjunction of two statements is true if at least one of the original
statements is true. Only one needs to be true in order for the conjunction to
be true.

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Conditional Statements

The most important way to combine two statements is by implication. The
implication of two statements *c* and *d* takes the form, "if *f*, then *g*." The result of implication is called a conditional statement. It is symbolized by placing an arrow between the two letters symbolizing the two statements, as so:

Figure %: The symbol for a conditional statement

Conditional statements don't necessarily imply cause and effect. They simply
state that if one event happens, then another will happen. Much of geometry can
be explained using conditional statements, and it is important to understand
them. For example, "if a polygon has three sides, then it is a triangle" is a
conditional statement.

A conditional statement has two parts, the hypothesis and the
conclusion. The hypothesis is the "if" clause of the statement. It is the
condition necessary for the conclusion to occur. The conclusion is the "then"
clause of the statement. The conclusion is true every time the hypothesis is
true. In the statement "If Julie runs fast, then she will win the race", the
hypothesis is "Julie runs fast" and the conclusion is "she will win the race."

Many different statements can be made by switching the hypothesis with the
conclusion and using the negation of a statement instead of the original
statement. In the next section, we'll look at some conditional statements with
their parts changed in certain ways, and we'll explore the truth values of such
statements.