Assignment on Discrete Mathematics

Added on - 24 Feb 2020

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DISCRETE MATHEMATICSQ1.Prove or disprove each of the following statements:(a)for all a,b belongs to z, if a congruent to b(mod 4)Then we have to check whether the average of a and b has the same parity as aFor instance let us take a=4 b=2 then the average of a and b is (a+b)/2=6/2=3 which is not same as aso the average of a and b does not have same parity as a that is this statement is disproved.(b) There exists n such that 4n^2-12n+8 is primeIf we take n=1 then 4n^2-12n+8=4-12+8=0If we take n=2 then 4n^2-12n+8=4(2)^2-12(2)+8=24-24=0If we take n=3 then 4n^2-12n+8=4(3)^2-12(3)+8=36-36+8=8If we taken=4 then 4n^2-12n+8=4(4)^2-12(4)+8=64-48+8=24Now we can find that 0,,0,8,24 are not prime’So the statement is disproved.(c)a, bZ, if 36a = 28b, then 7|a.If 36a=28b which implies 12a=7bIf a=14 then 7 divides a.(d) for all n belongs to z+check whether ( 4n^2+1)/n^2 =5If we take n=1 and n=2 we are not getting 5 as the answer.So the statement is disproved.(e ) for all n belongs to z+check whether (4n^2+1)/4=5If we check for instance n=1 we get the value as 5.If we take n=2 then we are getting 17/4So the statement is disproved(2)The reciprocal of a nonzero real number x is 1 x . Prove the following statement using a proof bycontradiction. The reciprocal of every irrational number is irrational.Proof:Let x be a irrational number.suppose 1/x is rational then 1/x=a/b where a and b are integers a notequal to 0 and since 0 is a rational number b not equal to 0X=1/1/x=1/a/b=b/a where b and a are integers and a not equal to 0[Type text]