Which of the following can be cited as a reason in a proof?a) Given c) Definitionb) Prove d) Postulate

Please answer all parts a) b) c) and d) A password is a string of 8 characters taken from the alphabet (that is, the set of 26 characters {a, b, c, ..., x, y, z}), the set of 10 digits {0, 1, 3, 4, 5, 6, 7, 8, 9} and the set of 5 special characters {!, @, #, ∗, +}. a) How many unique passwords are there? b) How many unique passwords are there where no character can be repeated and there are exactly 2 digits? c) How many unique passwords are there where you can repeat characters and there are at least 2 digits? d) How many unique passwords are there where you can repeat characters and have at least 2 digits and at least 1 special character?

prove the statement is true by using Mathematical Induction.

a) Which of the following statements are true? (a and b are integers.) Give informal ex- planations of your answers. i) If a and b are even, then ab is even. ii) If a and b are odd, then ab is odd. iii) If ab is odd, then a and b are both odd. iv) If ab is even, then a and b are both even. b) Formulate correct principles, similar in style to those of part i), for dealing with sums of odd and even integers.

Show work, keep simple.