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反证法是一种重要的证明方法。下面就几种类型的三角命题来阐述反证法的应用。一、对于命题的结论中出现“没有”、“不是”、“不能”之类的否定词,采用反证法是一种行之有效的方法例1 试证函数y=sinx~2不是周期函数。证明:假设y=sinx~2是周期函数,T>0是它的一个周期,则对任意实数有 sin(x+T)~2=sinx~2 令x=0,得sinT~2=sin0, 故sinT~2=0,∴T~2=kπ,又T>0, ∴T=(kπ)~(1/2) (k∈N) 令x=2~(1/2)T,得 sin(2~(1/2)T+T)~2=sin(2~(1/2))~2, sin〔2~(1/2)+1)~2kπ〕=sin2kπ, sin〔(2~(1、2)+1)~2kπ〕=0 ∴(2~(1/2)~2kπ=lπ (l∈N) (2~(1/2)+1)~2=l/k
Anti-evidence is an important method of proof. The application of counter-evidence is described below for several types of triangular propositions. First, there are negative words such as “no”, “no”, and “cannot” in the conclusion of the proposition. Using counter-evidence is an effective method. Example 1 The test function y=sinx~2 is not a periodic function. Proof: Suppose that y=sinx~2 is a periodic function and T>0 is a period of it, then for any real number there is sin(x+T)~2=sinx~2 Let x=0, get sinT~2=sin0, So sinT~2=0, ∴T~2=kπ, and T>0, ∴T=(kπ)~(1/2) (k∈N) Let x=2~(1/2)T, get sin (2~(1/2)T+T)~2=sin(2~(1/2))~2, sin[2~(1/2)+1)~2k[pi]]=sin2k[pi], sin[(2) ~(1,2)+1)~2kπ]=0 ∴(2~(1/2)~2kπ=lπ (l∈N) (2~(1/2)+1)~2=l/k