Question

(2 pts each) Find a different equivalent form of the statements. Justify your answers using Laws of equivalence or otherwise. (a) Not all men are Scientists. (b) If you are a computer science major you will need discrete mathematics.

Answer 1

1.

(a) Few men are not scientist too.

(b) Every computer science major needs discrete mathematics.

2. if the bit string of 10 bits does not start with 0 but end with 1

that means we fixed the first and last bit with 1.

then the num of such strings will be  $2^8=256$

3. given that,  $a_n=5a_{n-1}-6a_{n-2}$ with  

Now,

$a_n=5a_{n-1}-6a_{n-2}=2a_{n-1}+3a_{n-1}-6a_{n-2}=2a_{n-1}+3\left ( a_{n-1}-2a_{n-2} \right )$.............(1)

we have,  $a_1-2a_0=0$

then,  

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then  $\Rightarrow a_3-2a_2=0$ from (1)

$\Rightarrow a_4-2a_3=0$

similarly go on.

so we can say that,

$a_n-2a_{n-1}=0\Rightarrow a_n=2a_{n-1}$

Hence

$a_n=2^n\cdot a_0=2^n\cdot 1=2^n$