4.6 Indirect Argument: Contradiction and Contraposition “Reductio ad absurdum is one of a mathematician’s finest weapons. It is a far finer gambit than any chess gambit: a chess player may offer the sacrifice of a pawn or even a piece, but the mathematician offers the game." –G. H. Hardy, 1877-1947 “Reductio ad absurdum is one of a mathematician’s4.6 Indirect finestArgument: weapons. Contradiction It is a far andfiner Contraposition gambit than any chess gambit: a1 chess/ 7 player may offer the sacrifice of a pawn or even a piece, but the mathematician offers the game." –G. H. Hardy, 1877-1947 false." A Logical Defense “Suppose I did commit the crime. Then, at the time of the crime, I would have to be at the scene of the crime. In fact, at the time of the crime, I was in SM242, far from the scene of the crime, as the midshipmen will testify. This contradicts the assumption that I committed the crime, since it is impossible to be in two places at once. Hence the assumption is “Reductio ad absurdum is one of a mathematician’s4.6 Indirect finestArgument: weapons. Contradiction It is a far andfiner Contraposition gambit than any chess gambit: a2 chess/ 7 player may offer the sacrifice of a pawn or even a piece, but the mathematician offers the game." –G. H. Hardy, 1877-1947 A Logical Defense “Suppose I did commit the crime. Then, at the time of the crime, I would have to be at the scene of the crime. In fact, at the time of the crime, I was in SM242, far from the scene of the crime, as the midshipmen will testify. This contradicts the assumption that I committed the crime, since it is impossible to be in two places at once. Hence the assumption is false." “Reductio ad absurdum is one of a mathematician’s4.6 Indirect finestArgument: weapons. Contradiction It is a far andfiner Contraposition gambit than any chess gambit: a2 chess/ 7 player may offer the sacrifice of a pawn or even a piece, but the mathematician offers the game." –G. H. Hardy, 1877-1947 Suppose not. Then there exists an integer n that is even and odd. By definition of even, n = 2a for some integer a. By definition of odd, n = 2b + 1 for some integer b. Consequently, 1 2a = 2b + 1; and so a − b = : 2 1 Since a and b are integers, a − b is an integer. But a − b is 2 , which is not an integer. Thus a − b is an integer and a − b is not an integer, which is a contradiction. Method of Proof by Contradiction Procedure 1 Suppose the statement to be proved is false. (Suppose the negation of the statement is true.) 2 Show that this supposition leads to a logical contradiction. 3 Conclude that the statement to be proved is true. Theorem 4.6.2. There is no integer that is both even and odd. Proof: “Reductio ad absurdum is one of a mathematician’s4.6 Indirect finestArgument: weapons. Contradiction It is a far andfiner Contraposition gambit than any chess gambit: a3 chess/ 7 player may offer the sacrifice of a pawn or even a piece, but the mathematician offers the game." –G. H. Hardy, 1877-1947 Then there exists an integer n that is even and odd. By definition of even, n = 2a for some integer a. By definition of odd, n = 2b + 1 for some integer b. Consequently, 1 2a = 2b + 1; and so a − b = : 2 1 Since a and b are integers, a − b is an integer. But a − b is 2 , which is not an integer. Thus a − b is an integer and a − b is not an integer, which is a contradiction. Method of Proof by Contradiction Procedure 1 Suppose the statement to be proved is false. (Suppose the negation of the statement is true.) 2 Show that this supposition leads to a logical contradiction. 3 Conclude that the statement to be proved is true. Theorem 4.6.2. There is no integer that is both even and odd. Proof: Suppose not. “Reductio ad absurdum is one of a mathematician’s4.6 Indirect finestArgument: weapons. Contradiction It is a far andfiner Contraposition gambit than any chess gambit: a3 chess/ 7 player may offer the sacrifice of a pawn or even a piece, but the mathematician offers the game." –G. H. Hardy, 1877-1947 By definition of even, n = 2a for some integer a. By definition of odd, n = 2b + 1 for some integer b. Consequently, 1 2a = 2b + 1; and so a − b = : 2 1 Since a and b are integers, a − b is an integer. But a − b is 2 , which is not an integer. Thus a − b is an integer and a − b is not an integer, which is a contradiction. Method of Proof by Contradiction Procedure 1 Suppose the statement to be proved is false. (Suppose the negation of the statement is true.) 2 Show that this supposition leads to a logical contradiction. 3 Conclude that the statement to be proved is true. Theorem 4.6.2. There is no integer that is both even and odd. Proof: Suppose not. Then there exists an integer n that is even and odd. “Reductio ad absurdum is one of a mathematician’s4.6 Indirect finestArgument: weapons. Contradiction It is a far andfiner Contraposition gambit than any chess gambit: a3 chess/ 7 player may offer the sacrifice of a pawn or even a piece, but the mathematician offers the game." –G. H. Hardy, 1877-1947 contradiction. Method of Proof by Contradiction Procedure 1 Suppose the statement to be proved is false. (Suppose the negation of the statement is true.) 2 Show that this supposition leads to a logical contradiction. 3 Conclude that the statement to be proved is true. Theorem 4.6.2. There is no integer that is both even and odd. Proof: Suppose not. Then there exists an integer n that is even and odd. By definition of even, n = 2a for some integer a. By definition of odd, n = 2b + 1 for some integer b. Consequently, 1 2a = 2b + 1; and so a − b = : 2 1 Since a and b are integers, a − b is an integer. But a − b is 2 , which is not an integer. Thus a − b is an integer and a − b is not an integer, which is a “Reductio ad absurdum is one of a mathematician’s4.6 Indirect finestArgument: weapons. Contradiction It is a far andfiner Contraposition gambit than any chess gambit: a3 chess/ 7 player may offer the sacrifice of a pawn or even a piece, but the mathematician offers the game." –G. H. Hardy, 1877-1947 Method of Proof by Contradiction Procedure 1 Suppose the statement to be proved is false. (Suppose the negation of the statement is true.) 2 Show that this supposition leads to a logical contradiction. 3 Conclude that the statement to be proved is true. Theorem 4.6.2. There is no integer that is both even and odd. Proof: Suppose not. Then there exists an integer n that is even and odd. By definition of even, n = 2a for some integer a. By definition of odd, n = 2b + 1 for some integer b. Consequently, 1 2a = 2b + 1; and so a − b = : 2 1 Since a and b are integers, a − b is an integer. But a − b is 2 , which is not an integer. Thus a − b is an integer and a − b is not an integer, which is a contradiction. “Reductio ad absurdum is one of a mathematician’s4.6 Indirect finestArgument: weapons. Contradiction It is a far andfiner Contraposition gambit than any chess gambit: a3 chess/ 7 player may offer the sacrifice of a pawn or even a piece, but the mathematician offers the game." –G. H. Hardy, 1877-1947 Method of Proof by Contradiction Example Use the Proof by Contradiction method to prove that there is no largest integer. “Reductio ad absurdum is one of a mathematician’s4.6 Indirect finestArgument: weapons. Contradiction It is a far andfiner Contraposition gambit than any chess gambit: a4 chess/ 7 player may offer the sacrifice of a pawn or even a piece, but the mathematician offers the game." –G. H. Hardy, 1877-1947 If the animal is a sumo-wrestler, then it is not a duck. Argument Suppose an animal is a sumo-wrestler. Then it follows that this animal is a human being. Therefore the animal is not a duck. Statement to prove: There are no ducks that are sumo-wrestlers. Contrapositive Original: If the animal is a duck, then it is not a sumo-wrestler. Contrapositive: “Reductio ad absurdum is one of a mathematician’s4.6 Indirect finestArgument: weapons. Contradiction It is a far andfiner Contraposition gambit than any chess gambit: a5 chess/ 7 player may offer the sacrifice of a pawn or even a piece, but the mathematician offers the game." –G. H. Hardy, 1877-1947 Argument Suppose an animal is a sumo-wrestler. Then it follows that this animal is a human being. Therefore the animal is not a duck. Statement to prove: There are no ducks that are sumo-wrestlers. Contrapositive Original: If the animal is a duck, then it is not a sumo-wrestler. Contrapositive: If the animal is a sumo-wrestler, then it is not a duck. “Reductio ad absurdum is one of a mathematician’s4.6 Indirect finestArgument: weapons. Contradiction It is a far andfiner Contraposition gambit than any chess gambit: a5 chess/ 7 player may offer the sacrifice of a pawn or even a piece, but the mathematician offers the game." –G.