please  help to solve the problems as the attachment? The problemsisthe algorithms of computer science undergraduate level.  For example,  logic, sets sequence countings, intervals, Lexicographic-Order

1. Book problem #1: Prove that the product of any two odd integers is always an odd integer. 2. Book Problem #3: Construct a direct proof that for positive integers, if a|b and b|c, then a|c. 3. Book Problem #6: Disprove each of the following:
2
a. n +n+ 41 is prime for every positive integer n.
b. The product of any two irrational numbers is irrational.
c. The product of any rational number and any irrational number is irrational.
4. Book Problem # 9: Use Truth Tables to show that the following three Boolean expressions are equivalent:
a. (P ٨ Q) → R
b. P → (Q → R)
c. [P ٨ (~R)] → (~Q)
5. Book Problem #10: Is (P ٧ Q) → R equivalent to (P → R) ٧ (Q → R), or to (P→ R) ٨ (Q → R)?
2
n
6. Book Problem #14: Is n <3 Ɐ integers n ϵ ? If “no” give a counterexample, if “yes” give a proof by Mathematical Induction.
7. Book Problem #20: Find a formula for the sum below and use Mathematical Induction to show that
the formula is true Ɐ integers n ϵ .
1
1
1
1
+
+
+…+
(1)( 2) (2)( 3) (3 )( 4 )
n(n+ 1)
8. Book Problem #21a: Prove by Mathematical Induction that Ɐn ϵ it is true that:
n ∑ ( j+1)2 j=n 2n+1 +1
j=0

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