__is a rule that can be used in proving a theorem or some logic directly and polished.__

**A rule of inference**:**Addition:**is you just add another proposition onto it using the Logical OR.

p = 'The sky is blue"

p v q = "The sky is blue or red"

**Simplification:**is you just subtract a proposition

p ^ q = 2+2= 4 and grass is green

p = 2+2 = 4

**Conjunction:**adding two separate propositions together.

p = Grass is green.

q = Rome is in Italy

p^q = "Grass is green and Rome is in Italy".

**Modus Ponens:**Basically the end result of an implication.

P - > Q = "If I study, then i get an A".

Q = " I studied so i got an A"

**Modus Tollens:**Basically the opposite of Modus Ponens.

p - > q "If I go to the store, I buy milk"

~p = " I did not go to the store" Which means

~q = " I did not buy milk"

**Hypothetical Syllogism:**Say this out-loud to sound cool.Visually you can see how this works.

p -> q = "If i study, then i pass."

q - > r = " If i pass then i grad".

Therefore

p -> r = "If i study then i grad".

**Resolution:**

p v q = " I will study or eat dinner".

~p v r "I will not study or go to the party"

Therefore

q v r = " I will eat or go to the party"

**Disjunctive Syllogism:**

p v q = "I will study or I will eat"

~p = " I do not study"

q = "I will eat"

**Basic Proofs:**Proofs usually involve implications.

So two ways of solving these can be

**Contra-positive:**p -> q this is equivalent to ~q - > ~p. So instead of proving p - > q. we test ~p - > ~q to see if this is true. If it is true then we know p - > q is true.

Example:

If n + 2 is odd, then n is odd.

Suppose that "if n is even, then n + 2 is even"

let n = 2k for some integer k. n + 2 = 2k + 2 = 2(k+1). Clearly 2(k+1) is divisible by two thus making it even. Therefore since the contra-positive is true then p -> q is true.

**Contradiction:**This is a little more difficult. We want to prove that. p - > q is true and p -> ~q is true.

If both of them is true we go with the original as being true. For clearly both q and ~q cannot be true.

Example:

If n + 2 is odd, then n is odd.

Assume that n is odd. Then n = 2L + 1 for some integer L. So n + 2 = 2L + 3. Which is odd.

Suppose that If n + 2 is odd, then n is even.

Let n = 2k for some integer k. Then n + 2 = 2k + 2. which means n is even.

Since both is true, clearly if n + 2 is odd then n is odd is true.

oh man, good luck on the test, I know I wouldn't be able to pass it :x

ReplyDeleteMy friend is taking something like this in liberal arts, good luck man :D

ReplyDeletegood luck on the test

ReplyDeletethanks for the refreshing of some advanced mathematics

ReplyDeleteMan i love these kind of topics... i demand moar. [/ sarcasm-free comment]

ReplyDeleteI truly hate math, which is a shame because I love science.

ReplyDeleteknew all this, but it was a good reminder of stuff :) thanks

ReplyDeleteGot to be honest, while on holiday away from A level maths, I can't think about it... saw this and it just went "swoosh!" over my head lol. (I'll probably end up using it to revise at some point)

ReplyDelete