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There is one last point related to the justiﬁcation of the steps in a proof that is speciﬁc to this course in the foundations of geometry. We intend to build geometry on the real number system. Hence we will base many steps in our proofs on known facts about the real numbers. For example, if we have proved that x C z D y C z, we will want to conclude that x D y. Technically, this falls under the heading ‘‘by previous theorem,’’ but we will usually say something like ‘‘by algebra’’ when we bring in some fact about the algebra of real numbers.

This is a correct theorem. ) The contrapositive is this: If x2 Z 4, then however, is not correct. (x2 D 4 x Z 2. The contrapositive is a correct statement and is just a negative way of restating the original theorem. Consider another simple example: If x D 0, then x2 D 0. This time both the statement and its converse are true. Such statements are called biconditional statements and the phrase ‘‘if and only if’’ (abbreviated iff) is used to indicate that the implication goes both ways. In other words, P iff Q (or P 3 Q) means P * Q and Q * P.

They include such rules as the rules for negating compound statements that were described above and the rules for indirect proofs that will be described below. There is one last point related to the justiﬁcation of the steps in a proof that is speciﬁc to this course in the foundations of geometry. We intend to build geometry on the real number system. Hence we will base many steps in our proofs on known facts about the real numbers. For example, if we have proved that x C z D y C z, we will want to conclude that x D y.