Convert the following expression to cnf (conjunctive normal form) format using the propositional laws:

(p → q) → r

in your answer make sure that you (1) show your steps (2) do not skip any steps and (3) justify each step with the name of the law being used.

part(b) [50 points].

(i)(25 points) prove that the method of affirming (modus ponens) is sound using truth tables.

(ii)(25 points) prove that the method of denying (modustollens) is sound using truth tables.

See steps below

Step-by-step explanation:

a)

equivalence of (r implies s) with (not r or s)

De Morgan's Law

Double negation

Distributive Law

The last expression is in CNF.

b)

i)

Modus Ponens states the following,

If (p implies q) is true and p is true, then q is true.

By watching the truth table of implication

We can notice that the only row that satisfies  

(p implies q) is true and p is true

is the first row, so q must be true.

ii)

Modus Tollens states that if (p implies q) is true and (not q) is true, then (not p) is true.

By watching the following truth table

We can notice that the only row that satisfies (p implies q) is true and (not q) is true, is the fourth row, so (not p) must be true.

answer: haha (: i’ll go see what it is

step-by-step explanation:

3-4

step-by-step explanation:

the answer is 3-4 because both 45 and 60 has a multiple of 15.

45/3=15

60/4=15

therefore the odds against selecting a red is 4-3