1. Show by induction that: (1-x)(1+x+x 2 + x 3 +…+ x n = 1- x n-1 for n >=0 2. C
ID: 671083 • Letter: 1
Question
1. Show by induction that:
(1-x)(1+x+x2 + x3 +…+ xn = 1- xn-1 for n >=0
2. Consider the following argument:
If it is raining then the courts are closed
The courts are open
--------------------------------------------------------
.
. . It is not raining
Write this argument using propositional functions
Is this argument valid? Justify your answer.
3. Consider the set of strings on {0, 1} such that L(A) ={11wo| {0, 1}*}
Show that the language is regular.
4. Consider the following automaton:
Give its formal definition
Which of the following strings is accepted by the automation
Bb, a,
5. Consider the set X = {people in the world}, and the relation R ={ (x,y) iff x and y have the same age)
Is the relation reflexive, symmetric, anti-symmetric, transitive.
Explanation / Answer
1..-
The most basic geometric series is 1 + x + x2 + x3 + x4 + ... + xn. This is the finite geometric series because it has exactly n + 1 terms.
For n=1,
1 + x = {x^(1+1) - 1} / (x-1)
1+x = (x^2 - 1) / (x-1)
This is true
Assume for n=k it is true.
Check for n=k+1:
1 + x +x^2 ... x^k + x^k+1 = (x^(k+1) -1)/(x-1) + x^(k+1)
= { (x^(k+1) -1) + x^(k+1)*(x-1) } / (x-1)
= { x^(k+1) -1 + x^(k+2) - x^(k+1) } / (x-1)
= { x^(k+2) - 1 } / (x - 1)
= { x^((k+1)+1) - 1 } / (x - 1)
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