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 âàüa (read "not a").
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: 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.
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: 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.
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: 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.