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. H. Hardy, 1877-1947 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.

Argument Suppose an animal is a sumo-wrestler. Then it follows that this animal is a human being. Therefore the animal is 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: 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 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.

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. 

“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 Method of Proof by Contraposition

1 Express the statement to be proved in the form

∀x ∈ D, if P(x) then Q(x).

2 Rewrite this statement in the contrapositive form

∀x ∈ D, if Q(x) is false then P(x) is false.

3 Prove the contrapositive by a direct proof. 1 Suppose x is a (particular but arbitrarily chosen) element of D such that Q(x) is false. 2 Show that P(x) 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: a6 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 Contraposition

Example Prove the following statement by 1 contraposition and 2 contradition. For all integers n, if n2 is odd, then n is 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: a7 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