双语离散计算ch2

OR:

show that the contrapositive (~q)(~p) is true.
Since (~q) (~p) is logically equivalent to p q, then the theorem is proved.
Proof by contrapositive

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1.1 Mathematical Systems, Direct Proofs, and Counterexamples

A mathematical system consists of

Undefined terms Definitions Axioms
Undefined terms

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Example 2.1.11(13): Given a direct proof of the following statement: For all X, Y, and Z (1) XÇ(Y-Z)=(XÇY)-(XÇZ); (2) XÈ(Y-X)=XÈY.
Discussion: Proof: (1)Let X,Y, Z be arbitrary sets. We prove XÇ(Y-Z)=(XÇY)-(XÇZ) by proving (i) if xÎ XÇ(Y-Z), then xÎ (XÇY)-(XÇZ) and (ii) if xÎ (XÇY)-(XÇZ) , then xÎXÇ(Y-Z)
Theorems

A theorem is a proposition of the form p q which must be shown to be true by a sequence of logical steps that assume that p is true, and use definitions, axioms and previously proven theorems.

Theorem are often of the form For all x1,x2,…,xn , if p(x1,x2,…,xn) then q(x1,x2,…,xn) A direct proof assumes that p(x1,x2,…,xn) is true and then using p(x1,x2,…,xn) as well as other axioms, definitions, previously derived theorems, and rules of inference, shows directly that p(x1,x2,…,xn) is true Definition 2.1.7 An integer n is even if there exists an integer k such that n=2k. An integer n is odd if there exists an integer k such that n=2k+1



Two triangles are congruent if their vertices can be paired so that the corresponding sides are equal and so are the corresponding angles. he sum of their measures is 180 degrees.

….
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Disproving a Universally Quantified Statement

Recall: to disprove "xP(x)
we need to find one member x makes P(x) is false. Such a value for x is called a counterexample.
Point Line

Definitions
A definition is a proposition constructed from undefined terms and previously accepted concepts in order to create a new concept. Example. In Euclidean geometry the following are definitions:

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Indirect proof
The
method of proof by contradiction of a theorem p q consists of the following steps:
1. Assume p is true and q is false 2. Show that ~p is also true. 3. Then we have that p ^ (~p) is true. 4. But this is impossible, since the statement p ^ (~p) is always false. There is a contradiction! 5. So, q cannot be false and therefore it is true.
Lemmas and corollaries
A lemma is a small theorem which is used to prove a bigger theorem. A corollary is a theorem that can be proven to be a logical consequence of another theorem. Example from Euclidean geometry: "If the three sides of a triangle have equal length, then its angles also have equal measure."


Given two distinct points, there is exactly one line that contains them. Given a line and a point not on the line, there is exactly one line through the point which is parallel to the line.

The method of proof by contrapositive of a theorem p q is to show that (~q)(~p) is true.

Since (~q) (~p) is logically equivalent to p q, then the theorem is proved.
Proof by Contradiction A proof by contraction establishes pq by assuming that the hypothesis p is true and that the conclusion q is false and then, using p and q as well as other axioms, definitions, previously derived theorems, and rules of inference, derives a contradiction. A proof by contradiction is sometimes called an indirect proof
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Example 2.1.8: The integer n=12 is even because there exists an integer k=6 such that n=12=2k=2´6
Example 2.1.10: Given a direct proof of the following statement: For all integers m and n, if m is odd and n is even, then m+n is odd. Discussion: Proof: Let m and n be arbitrary integers, suppose that m is odd and n is even. Then, by definition, there exists two integers k1 and k2 such that m=2k1+1, n=2k2 . Then, the sum is m+n= (2k1+1)+2k2 =2(k1+k2 )+1. Thus, there exists an integer k=k1+k2 such that m+n=2k+1. Therefore m+n is odd.
Axioms
An axiom is a proposition accepted as true without proof within the mathematical system. There are many examples of axioms in mathematics: Example: In Euclidean geometry the following are axioms

Types of proof

A proof is a logical argument that consists of a series of steps using propositions in such a way that the truth of the theorem is established. Direct proof: p q