Let S = {a1,a2,a3,...an}be a set of test scores. Prove using the the indirect method of proof that if the average of this set of test scores is greater than 90, then at least one of the scores is greater than 90.

We can use contrapositive rule to prove it,

for example $P\rightarrow Q$ ,the contrapostive of this will be ~Q$\rightarrow$ ~P

we can covert " if the average of this set of test scores is greater than 90, then at least one of the scores is greater than 90 "into proposition logic $P\rightarrow Q$

contrapositive of this will be (  ~Q$\rightarrow$ ~P) both are equal

so we can prove it through contrapositive rule "none of them scores is greater than 90 then the average of this set of test scores is not greater than 90" means $a_1<90 , a_2<90 , a_3<90...........a_n<90$ (none of them greater than 90)

then sum of these will be $a_1+a_2+a_3.............a_n<90n$

and the average will be ${a_1+a_2+a_3.............+a_n\over n }= {90n\over n}$

hence the average will be les than 90