- Please I need help and advice in how to answer the problems below.

(a) Below are two attempts at deductions. In each case, suppose that we know that the first two propositions are true, and wish to deduce that the third is true. In each of parts (i) and (ii), state whether the deduction is valid. If it is valid, give a deduction combining proof by contradiction and Modus Ponens. If it is not valid, explain why not.

(i) We know that:

If I have my car keys, then I can open my car door.I cannot open my car door.

We conclude that:

I do not have my car keys.

(ii) We know that:

If I have my car keys, then I can open my car door.I can open my car door.

We conclude that:

I have my car keys.

(b) (i)Show that for any positive integer n, n^4 −n^2 = n × (n−1)n(n+1)

(ii) Deduce that n^4 − n^2 + 1 always has remainder 1 upon division by 3, for any positive integer n.