非正式课程测试功能逻辑-溶液functionslogic solutions表达.pdfVIP

InformalClassTest:FunctionsandLogic-solutions

Functions

()7−1()−1

(a)Letℝ→ℝbegivenbyf=.Findand().

34

Answer:

−1(=4x+7and−1()=x

3x−1

()()

(b)Letℝ→ℝbegivenby=3x+4.Provethatisabijectivefunction.

Answer:

()

isonetooneandonto,because:

1.onetoone:itisnothardtoseethatf(x)=f(y)whichgivesx=y.

2.onto:lety=f(x)whereyisintherangeandfindanxinthedomain.Transposingit,weget

x=(y-4)/3.Sinceyisarealnumber,xisalsoarealnumber.Onetoone+ontoimpliesfisa

bijectivefunction.

(c)CSE107has60students.Provethatthereareatleasttwostudentswiththeirfirstnamestarting

withthesameletter.

Answer:

Thereare60students(pigeons)andonly26letters(holes).BythePHP,wecanfinethatsometwo

studentswithfirstnamestartingwiththesameletter.

Logic