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CHAPTER 14 Improvement of Reasoning
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14.3 Converse, Inverse, and Contrapositive of a Statement
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DEFINITION 1:
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The converse of a statement is the statement that is formed by interchanging the hypothesis and conclusion.
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Thus, the converse of the statement lions are wild animals is wild animals are lions. Note that the converse is not necessarily true.
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The negative of a statement is the denial of the statement.
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Thus, the negative of the statement a burglar is a criminal is a burglar is not a criminal.
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The inverse of a statement is formed by denying both the hypothesis and the conclusion.
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Thus, the inverse of the statement a burglar is a criminal is a person who is not a burglar is not a criminal. Note that the inverse is not necessarily true.
DEFINITION 4:
The contrapositive of a statement is formed by interchanging the negative of the hypothesis with the negative of the conclusion. Hence, the contrapositive is the converse of the inverse and the inverse of the converse.
Thus, the contrapositive of the statement if you live in New York City, then you will live in New York State is if you do not live in New York State, then you do not live in New York City. Note that both statements are true.
14.3A Converse, Inverse, and Contrapositive Principles
PRINCIPLE PRINCIPLE
1: 2:
A statement is considered false if one false instance of the statement exists. The converse of a definition is true.
Thus, the definition a quadrilateral is a four-sided polygon and its converse a four-sided polygon is a quadrilateral are both true.
PRINCIPLE
The converse of a true statement other than a definition is not necessarily true.
The statement vertical angles are congruent angles is true, but its converse, congruent angles are vertical angles is not necessarily true.
PRINCIPLE
The inverse of a true statement is not necessarily true.
The statement a square is a quadrilateral is true, but its inverse, a non-square is not a quadrilateral, is not necessarily true.
PRINCIPLE
The contrapositive of a true statement is true, and the contrapositive of a false statement is false.
The statement a triangle is a square is false, and its contrapositive, a non-square is not a triangle, is also false. The statement right angles are congruent angles is true, and its contrapositive, angles that are not congruent are not right angles, is also true.
14.3B Logically Equivalent Statements
Logically equivalent statements are pairs of related statements that are either both true or both false. Thus according to Principle 5, a statement and its contrapositive are logically equivalent statements. Also, the converse and inverse of a statement are logically equivalent, since each is the contrapositive of the other. The relationships among a statement and its inverse, converse, and contrapositive are summed up in the rectangle of logical equivalency in Fig. 14-3:
CHAPTER 14 Improvement of Reasoning
Fig. 14-3
1. Logically equivalent statements are at diagonally opposite vertices. Thus, the logically equivalent pairs of statements are (a) a statement and its contrapositive, and (b) the inverse and converse of the same statement. 2. Statements that are not logically equivalent are at adjacent vertices. Thus, pairs of statements that are not logically equivalent are (a) a statement and its inverse, (b) a statement and its converse, (c) the converse and contrapositive of the same statement, and (d) the inverse and contrapositive of the same statement.
SOLVED PROBLEMS
Converse of a statement State the converse of each of the following statements, and indicate whether or not it is true. (a) Supplementary angles are two angles the sum of whose measures is 180 . (b) A square is a parallelogram with a right angle. (c) A regular polygon is an equilateral and equiangular polygon.
Solutions
(a) Two angles the sum of whose measures is 180 are supplementary. (True) (b) A parallelogram with a right angle is a square. (False) (c) An equilateral and equiangular polygon is a regular polygon. (True)
Negative of a statement State the negative of (a) a b; (b) m B m C; (c) does not lie on the line.
Solutions
(a) a b (b) m B (c) m C D.
C is the complement of
D; (d) the point
C is not the complement of
(d) The point lies on the line.
Inverse of a statement State the inverse of each of the following statements, and indicate whether or not it is true. (a) A person born in the United States is a citizen of the United States. (b) A sculptor is a talented person. (c) A triangle is a polygon.
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