Ampqp qdiscrete math logic transcriptokay

The �rst form is called Propositions. So what are propositions? A proposition is a statement, or another word is a sentence. So it is a statement or a sentence, but this statement or a sentence. It has to have a condition which is we can determine whether it is true or false. So it is a statement or a sentence that can be
determined to be either true or false. Let us take some examples here.

First the example is "John is a programmer." Based on our knowledge about John, we know that this statement can be true, or it can be false, depending on our knowledge about John. Another example will be something like, "I wish I were wise." So I wish I were wise cannot be determined to be true or false, because it is simply a wish. So John is a programmer is in fact a proposition; I wish I were wise, this is not a proposition. So to summarize a proposition is a statement or a sentence that can be determined to be either true or false.

(AND)

Inclusive ∨

Negation ¬
(NOT)

Implication →
Double ↔︎
Implication

disjunction OR and the symbol for it is the small ∨. The third one and the most common connectors is called Exclusive disjunction OR. So we have Inclusive disjunction OR and Exclusive disjunction OR and we are going to talk about both of them in details and see the differences. Now the symbol for it is the . We also⊻

have the Negation, and the symbol for this is the ¬

Truth Table of Conjunction, AND

So and can both be true, can be true is false, then is false is true, �nally

p ∧ q is true only when is true and is true.

Truth Table of Disjunction,

Truth Table of Exclusive

Disjunction, XOR

Now, we are going to talk about the inclusive disjunction. So, or is false, we are talking about inclusive disjunction . Inclusive disjunction is false only when both and are false.

So, both of them have to be false at the same time. All right, let us construct this truth table and discuss it in more detail. So, here we have our , our and p ∨ q . And we saw in our previous slides that we can discuss every possible case here. So, and are: both true, true false, false true, and false false.

true but not both.

Okay here is the easiest one ever: Negation. Now negation of . The negation of p is the opposite of or we call it ¬p . So, ¬p is false when is true and ¬p is true when is false. So basically this is just one proposition here. We are not dealing with , just .

So, basically this column right there. I am going to frame it in grey. The purpose for it is to simply it. So, I am taking it one step at a time. All right, let us discuss all of the different options.

All right, so here we have three parameters. Every parameter can have two different cases. So, we determine it by two to the power of three, which are eight different cases. Why truth to the power of three? Every proposition , , or OR can p q
be true or false so two values, binomial values, and there are three propositions so that is the exponent will be eight.

All right. Now, we are concern about the grey column and the purple column, the OR to be able to get the last one, to get the �nal result.

So, the or and OR. or and R would only be true if both are true. So, here true and true would be true, let us see again here it is true and true would be true, true and true is true and that is it. Otherwise it is false. So, here we have a false, false, false, false, false, as you can see here I am only studying based on the limiting factor, when I do the end, when I do the conjunction I am thinking both have to be true. Otherwise, it does not matter. Otherwise, if one of them is false or both are false then the outcome is false.

For example: if is "John is a programmer", and is "Mary is a lawyer", then p condition states that if John is a programmer then Mary is a lawyer. Of course, the statement can be true or false. We do not know that. In this case here, if that or not really be accountable with each other, so John can be a programmer while Mary can be another programmer or an engineer, she does not have to be a lawyer. So here we are not judging whether this statement is true or false; we are just giving the example of the dependability or conditional proposition.

right, so let us draw this. We have true true, true false, false true, and false false. The same conditions we have inside. Now, p → q is true when both and are true. So when and are true, then the outcome is true. So here we have that AND relationship. Or another totally different condition: when is false. So is

Logical Equivalence

All right, a way to discuss logical equivalence. Now what is logical equivalence? Two prepositions are set to be logically equivalent if the truth tables are identical.

¬p is the opposite of . So all we have to worry about it is the �st column here and do the exact opposite. So, false false, true true, and then we discuss ¬p ∨ q , the inclusive disjunction. So here, it will only be false if both are false. Remember we discussed the truth table for the inclusive disjunction? Otherwise it is going to be true; the statements are going to be true. All right, well that was easy.

Converse

Alright, now we can discuss what is called the Converse. So the Converse of

Alright, now let us do the last one here. The last one is we have to imagine as if we reverse the �rst two columns. So if both are true, then the outcome is in fact true. Otherwise if is false, then the outcome is true regardless of the value of . Otherwise it is false here.

So from this we conclude that the two propositions are not logically equivalent. There is no logically equivalent �uency relationship here because the truth tables are not identical.

Alright, another important concept to discuss is the Contrapositive. So the

is the contrapositive.

Truth table for double implication of statement and state-ment q.

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Truth table for a Tautology po
tion for statement and stat
ment q.

So, if both are true then the outcome is true or if the �rst one is false, regardless of the second one, the answer is true, because basically all of them are true, then we can say that this proposition is in fact a Tautology.

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Alright, well the only way to prove this is by truth table. Like anything in logic we'd probably be using truth tables. So let us create the truth tables. Here we have p and , then p ∨ q , then we have the ¬(p ∨ q) , how about the ¬p and ¬q , and�nally the last one is (¬p) ∧ (¬q) . Alright, as we have done in the past, we can just determine all the different options there. So true true, true false, false true, false false.

Okay p ∨ q means that it will only be false if both are false. Otherwise it is true. If one or both are true it is true. Then doing them ¬(p ∨ q) will be very easy because all we have to do is to reverse the third column. So this will be false, false, false, true. Now if we do the ¬p , the opposite of will be false, false, true, true. ¬q will be false, true, false, true. Then the (¬p) ∧ (¬q) ) will be false if any one of them is false, and will be true only if both are true. So from this here we can conclude that (¬p) ∧ (¬q) is logically equal to ¬(p ∨ q) .

false, false true, false false.

Now all we need to in there is to look at the last two columns and do the OR relationship. So the OR relationship states that if one of them or both are true then the outcome is true. So true, true, true, and the �rst one is false, false, so it is false.

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