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Discrete StructuresIntroduction to Proofs

Dr. Muhammad HumayounAssistant Professor

COMSATS Institute of Computer Science, Lahore.mhumayoun@ciitlahore.edu.pk

https://sites.google.com/a/ciitlahore.edu.pk/dstruct/Some material is taken from Dr. Atif’s slides

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Terminology• Theorem: a statement that can be shown true.

Sometimes called facts.• Proof: Demonstration that a theorem is true.• Axiom: A statement that is assumed to be true.• Lemma: a less important theorem that is useful

to prove a theorem.• Corollary: a theorem that can be proven directly

from a theorem that has been proved.• Conjecture: a statement that is being proposed to

be a true statement.

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Stating Theorems

• Theorem. If , where x and y are positive real numbers, then .

• Theorem. For all positive real numbers x and y, if , then .

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Methods of Proving Theorems

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Trivial Proofs

• Consider an implication:

• If it can be shown that p is true, then the implication is always true– By definition of an implication

• Note that you are showing that the conclusion is true

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Trivial Proof Example

Consider the statement:• If you are in CSC102 then you are a student.

• Since all people in CSC102 are students, the implication is true.

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Vacuous proofs

• Consider an implication:

• If it can be shown that is false, then the implication is always true.– By definition of an implication

• Note that you are showing that the hypothesis is false

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Example

Consider the statement:• Every student snooze during class when he is

tired.• Rephrased: If a student is tired then he snoozes

during class

• Since there is no such student who is snoozing, (the hypothesis is false), the implication is true

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Methods of Proving Theorems

Important ones

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Methods of Proving Theorems

Direct Proofs• Consider an implication: – If p is false, then the implication is always true– Thus, show that if p is true, then q is true

• To perform a direct proof, assume that p is true, and show that q must therefore be true

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Example Direct Proof

Theorem. Show that the square of an even number is an even number.For every number , if is even, then is even.

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Example Direct Proof

Theorem. Show that the square of an even number is an even number.For every number , if is even, then is even.

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Example Direct Proof

Theorem. Show that the square of an even number is an even number.For every number , if is even, then is even.

Proof. Usual convention: Universal instantiation is not explicitly used and it is directly showed that implies .

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Example Direct Proof

Theorem. Show that the square of an even number is an even number.For every number , if is even, then is even.

Proof. Usual convention: Universal instantiation is not explicitly used and it is directly showed that implies .

Assume is even.Thus, , for some (definition of even numbers).

As is 2 times an integer, is thus even.

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Example Direct Proof

Theorem. Show that the square of an even number is an even number.For every number , if is even, then is even.

Proof. Usual convention: Universal instantiation is not explicitly used and it is directly showed that implies .

Assume is even.Thus, , for some (definition of even numbers).

As is 2 times an integer, is thus even.

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Proof by Contraposition(or Indirect proofs)

– If the antecedent is false, then the contrapositive is always true

– Thus, show that if is true, then is true

• To perform an indirect proof, do a direct proof on the contrapositive

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Indirect proof exampleTheorem. If is an odd integer then is an odd integerProof (by contrapositive). We show that the contrapositive of the theorem statement holds, therefore the theorem statement hold. Contrapositive: If is an even integer, then is an even integer.

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Indirect proof exampleTheorem. If is an odd integer then is an odd integerProof (by contrapositive). We show that the contrapositive of the theorem statement holds, therefore the theorem statement hold. Contrapositive: If is an even integer, then is an even integer.Assume that is even. Then by the definition of even numbers: for some integer .

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Indirect proof exampleTheorem. If is an odd integer then is an odd integerProof (by contrapositive). We show that the contrapositive of the theorem statement holds, therefore the theorem statement hold. Contrapositive: If is an even integer, then is an even integer.Assume that is even. Then by the definition of even numbers: for some integer . Thus

Since is 2 times an integer, it is even. Hence our proof by contraposition succeeds.

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Selecting a Proof method

Theorem. If is an integer and is odd, then is even.Proof. (Via direct proof). Assume that is odd. Then by the definition of odd numbers for some integer k. Thus

… ???Direct proof doesn’t seem to work.

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Selecting a Proof method

Theorem. If is an integer and is odd, then is even.Proof. (Via direct proof). Assume that is odd. Then by the definition of odd numbers for some integer k. Thus

… ???Direct proof doesn’t seem to work.

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Selecting a Proof methodTheorem. If is an integer and is odd, then is even.

Indirect proof. Contrapositive: If is odd, then is even.Assume is odd, and show that is even.By the definition of odd numbers for some integer .

As is times an integer, it is even

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Selecting a Proof methodTheorem. If is an integer and is odd, then is even.

Indirect proof. Contrapositive: If is odd, then is even.Assume is odd, and show that is even.By the definition of odd numbers for some integer .

As is times an integer, it is even

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Selecting a Proof methodTheorem. If is an integer and is odd, then is even.

Indirect proof. Contrapositive: If is odd, then is even.Assume is odd, and show that is even.By the definition of odd numbers for some integer .

As is times an integer, it is even

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Selecting a Proof methodTheorem. If is an integer and is odd, then is even.

Indirect proof. Contrapositive: If is odd, then is even.Assume is odd, and show that is even.By the definition of odd numbers for some integer .

As is times an integer, it is even

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Proof by contradiction (another type of indirect proofs)

• Given a statement of the form

• Assume p is true and q is false–Assume

• Then prove that cannot occur–A contradiction exists

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Proof by contradiction exampleTheorem. If is an integer and is odd, then is even.Rephrased: If is odd, then is evenProof. Assume p is true and q is false (Assume . Assume )

Assume that is odd, and is odd.

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Proof by contradiction exampleTheorem. If is an integer and is odd, then is even.Rephrased: If is odd, then is evenProof. Assume p is true and q is false (Assume . Assume )

Assume that is odd, and is odd.By the definition of odd numbers for some integer k.

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Proof by contradiction exampleTheorem. If is an integer and is odd, then is even.Rephrased: If is odd, then is evenProof. Assume p is true and q is false (Assume . Assume )

Assume that is odd, and is odd.By the definition of odd numbers for some integer k.

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Proof by contradiction exampleTheorem. If is an integer and is odd, then is even.Rephrased: If is odd, then is evenProof. Assume p is true and q is false (Assume . Assume )Assume that is odd, and is odd.By the definition of odd numbers for some integer k.

As is 2 times an integer, it must be even.Contradiction!

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End