Discrete Mathematics Answers

Define the following with an example

(I) tautology

(I) proposition

In the following argument, determine the validity or otherwise of the statement:
"If the BIOS test runs fine, the CPU and motherboard must be OK. If the CPU and the motherboard and memory are all OK, then there must be a flaw in the OS. The BIOS test runs fine and the memory is OK. Therefore there must be a flaw in the OS."

(I) Write the four(4) propositional statement(s) in the above argument.

(II) State the premise(s) and conclusion in the above argument.

(III) Using a truth table, determine the validity or otherwise of this statement.

Prove by induction that
1+2+3+.....+n = n(n+1)÷2

Give a proof of the theorem, "if n is odd, then n squared is odd".

By constructing truth tables, decide which of the following are tautologies.

(I) ~(P ^ ~P)

(II) P implies ~P

(III) (P ^ (p implies q)) implies q

B) show that (P implies Q) implies R is logically equivalent to
(~ P implies R) ^ (Q implies R)

A) Let P be the propositions Roses are Red and Q be the propositions Violets are Blue.
Express each of the following propositions as logical expressions:

(I) If roses are not red, then violets are not blue.

(II) Roses are Red or Violets are not blue.

(III) Either Roses are Red or Violets are Blue (but not both)

Identify the error or errors in this argument that supposedly shows that if
∃xP (x) ∧ ∃xQ(x) is true then ∃x(P (x) ∧ Q(x)) is true. ∧