17 de julio de 2026

Good morning. Today we are going to start with our asynchronous classes of the subject of Algebra. We will start with the topic number 1, propositional logic. Here we have a table with the respective symbols that are used in propositional logic. And we also have on the right side how each of those symbols is read.

En este primer símbolo vemos que su lectura o cómo se lee es no o negación de. En el segundo símbolo tenemos este y se lee y. En el tercer símbolo vemos que se lee o. El cuarto que es como una flecha se lee entonces. El quinto símbolo es una flecha de doble sentido se lee si y solo si. Y el último símbolo que es como una ve.

with a line below. It is read or it is one or it is the other but not both. If this reading seems very long to you, the other option is simply to say: difference, symmetrical. With that clarification, let's see the true tables. We have the negation, the conjunction, inclusive disjunction, implication, double implication, exclusive disjunction. In this course,

We are going to use as values for each proposition. The number 1 and the number 0. 0 means off and 1 means on. As you can see here we have two images of two little lights. One of them is off and the other is on. It may be that in other courses you have learned with false and true.

The off is known as false and the on is known as true. But in this course we are going to use as I have mentioned the values 1 and 0, an on and off. I already sent the task number 1 where you can also see each of the true tables and in turn the classification of the propositional formulas.

Tautology is when the entire resulting column is 1. Contradiction is when the entire resulting column is 0. Contingency is when we have different values. They can be 1s or 0s. And that is called contingency. To solve the exercises, you must use the values 1 and 0. Now, to make it better understood,

Vamos a resolver un par de ejercicios que también estaban incluidos en sus tareas. Tenemos un primer ejercicio en el cual tenemos esta fórmula proposicional. En toda esta fórmula proposicional, conocida también como proposición compuesta, tenemos únicamente dos proposiciones simples que son la P y la Q. En toda la fórmula proposicional, como son solamente dos, vamos a tener...

Cuatro filas, primera fila, segunda fila, tercera fila, cuarta fila. La primera proposición que espe tendrá cuatro valores. Dos encendidos y dos apagados. La segunda proposición es Q. Tendrá un valor encendido. Otro apagado. Nuevamente un valor encendido y apagado. Con estos valores vamos a llenar a todas las proposiciones simples que tenemos aquí.

Here is the P. It would be two on, two off. But if you notice, in front of that P there is a negation that would change its values. It would be two off, two on. Here we have another P. It would be two on, and two off. Here is another P. Two on, two off. Here we also have a P. Two on, two off.

and we don't have more p's, but this one here has a negation ahead that would change its values. Turned off. On. Then, the next proposition would be the Q, the one that has a on/off. We write a on/off. Turned off again, but it has a negation ahead that would change. What would change its values?

Apagado encendido, apagado encendido. Aquí también tenemos una Q. Sus valores serían encendido apagado, encendido apagado. Aquí también encendido apagado, encendido apagado. Finalmente tenemos esta Q. Encendido apagado, encendido apagado. Con eso hemos completado de asignarle sus respectivos valores a cada proposición simple. Es decir a cada letra.

Now we have to find the values of the grouping signs. We start with the parentheses, that is, we need the result of this. We are going to mark it with a color so that it can be different from the rest. There we have parentheses and we would have to use the negation of Q and P, that is, this column and this other one, so that the result is

"Encendido" The two should be on as in this case. We are not going to consider this column. We will put it of a color with which it is not very noticeable. There. What interests us is this column and this other one. With this symbol that is read and if the two are on the result is on. The rest would be off. That way we already have the result.

From parentheses, we will focus on that result. That is, in this column. We go out of the parentheses. And now we have other grouping signs that would be the corchets. All this, we would have to connect this negative column of P with this other column that I have also marked, of yellow color. Because everything is inside the corchets. All that part.

That is, all of that is inside the brackets. The two columns that are yellow are the ones that have a connection with this symbol that is read then. You set your tables for real. That corresponds to the implication and the result is off when the antecedent is on and the consequent is off. We are looking for a case like this.

Antecedente encendido. Consecuente apagado. Resultado apagado. Entre estas dos columnas de color amarillo. Antecedente encendido. Consecuencia apagado resultado. Apagado. Antecedente encendido. Consecuente apagado. Resultado apagado. Y en estos otros dos casos pues. El resultado sería encendido. Apagado entonces. Apagado. Resultado encendido.

We look at the table. Turned off. Then turned off, result on. And we have written it down. Here. The same here turned off. Then on, result on. This is the result that is inside the corks. This green column is the result of these corks. But in front.

of the corks we have a negation, that is, here would be that result of the corks denied. This column would change by this negation, it would become off, off, on, on. With that, we would just have the result of the whole cork. That result has a connection with the values ​​of this P.

is connected to this symbol that is read with the "o" enough that one of the values is turned on so that our result is also turned on. We will look at these two columns, that and this other are connected with the "o" with the "o" enough that one of the two is turned on so that the result is also turned on. As you see, the whole column

gives as a result turned on and so we have our result of everything that was inside the keys look everything that was inside the keys has given us a result that is this we are going to put it of a color with which it can be differentiated there it is and well we can remove the color to this so that it visualizes the final result what would this column be

Todo está encendido. Con eso tenemos el resultado de toda esta parte que está dentro de las llaves. Vamos al otro lado, es decir, aquí. Empezamos dentro de los paréntesis aquí. Tenemos la I. Con la I los dos tienen que estar encendidos para que el resultado también salga encendido, es decir, uno. Lo demás sería apagado, apagado, apagado.

With that we have the result of the parentheses. But in front of the parentheses there is a negation. That negation would change the values ​​that were inside the parentheses. And well, there we would have to change this by the negation. On, on, off, on. This result corresponds to the parentheses. Later this column.

se conecta con esta otra mediante este símbolo que se lee entonces. Con ese símbolo, ¿qué es la implicación? Nos da apagado. Cuando el antecedente está encendido y el consecuente apagado. Busquemos ese caso. Antecedente encendido. Consecuente apagado resultado. Apagado y el resto estaría encendido encendido. Encendido. Aquí también tendríamos

a value of "off" because the antecedent is on, but the consequent is off. "off result" This column is the result of what was inside the brackets. That is, we can remove the color. The same here. So we focus only on this column. Now,

Fuera de los corchetes todavía tenemos Q entonces. Es decir esto. Por lo cual este es resultado de Q. Por tanto, lo que tendríamos que hacer es conectar esta columna con esta otra. Nos fijamos que con qué símbolo están conectados. El símbolo es el entonces. Con ese símbolo nos da apagado. Cuando el antecedente está encendido y el consecuente apagado, miren la tabla.

Antecedente encendido consecuente apagado. Resultado apagado. Busquemos ese caso porque todos los demás da como resultado encendido. No hay mucho que analizar. Busquemos un antecedente, uno y un consecuente cero en otras palabras. Antecedente encendido consecuente apagado. Resultado apagado. Antecedente consecuente. Y vemos que no hay un caso como ese.

that the antecedent is on and the consequent is off. Since we don't have that case, it means that the whole result of this column with the conditional would be on. This column is the result of everything that was inside the keys and we can remove the color to the rest. To focus only on the column that is painted.

We said that that column is the result of these keys. But in front of those keys, we have a negation. Therefore, all this changes. Turned off, turned off, turned off, turned off. We are going to remove the color of all this. What we are interested in is the final result of everything that was inside the keys.

Su resultado final está en la negación que lo teníamos delante de la llave. Con eso tenemos el resultado final de estas llaves y el resultado final de estas otras llaves. Y al medio tenemos este símbolo que es la doble implicación. Con la doble implicación, los dos tendrían que estar iguales. Si esto es encendido,

This one should also be turned on, but since they are not the same, the result would be the same here, here and here. This column is the final result. So, we will remove the color that is already there, which does not represent much to know the real value of this propositional formula. Our resulting column is this one as everything came out.

0 0 0 is if everything is off. We would have to put the result here. What would it be? Contradiction. Because everything is 0. Well, let's go with the next exercise. What would exercise 2 be? Don't forget that these classes are asynchronous. They are programmed classes. With anticipation. So that later they activate. And you can see them.

We have a second exercise, this one, where our compositional propositional formula has three simple propositions. P, Q, R. When that happens, when you have three simple propositions within a whole compositional propositional formula, there will be eight rows. One row is this, the other row, third.

Cuarta, quinta, sexta, séptima, octava fila. Eso ocurre cuando tienes tres proposiciones. En este otro ejercicio teníamos solo dos. Y por eso había cuatro filas. Pero como aquí tenemos tres, tendríamos como resultado ocho filas. Asignamos valores. Primeramente a la P. Tendría valores de encendido. Cuatro encendidos y cuatro apagados. Luego la Q.

It would be two on, two off. We complete repeating the same. Two on, two off. Finally, the R would have on off, on off, on off, on off. With all that, we begin to assign the values to this entire table in which our propositional formula is composed. Let's start with P.

Encendido, encendido, encendido. Luego, apagado, apagado, apagado, apagado. Pero tiene una negación por delante entonces cambiarán sus valores. Apagado, apagado, apagado. Apagado, luego, encendido, encendido, encendido, encendido. Listo. Buscamos otra P. Aquí está. Sería encendido cuatro veces. Apagado cuatro veces.

y ya no tenemos más p. Lo que viene después es la q. Esta que tendría dos encendidos, dos apagados y repetimos dos encendidos, dos apagados. Ya no hay más q. Lo que sí tenemos es la r. La r tiene un encendido, un apagado y se repite.

"Encendido apagado" "Encendido apagado" "Encendido apagado" But this "R" has a negation in front of it. They would change their values. It would be "Apagado encendido" "Apagado encendido" "Apagado encendido" "Apagado encendido" These two columns where the negations are. That we are going to paint here to guide us. Those two columns.

belong to everything that was within these parentheses and its final result would be here in the middle with the symmetrical difference in the symmetrical difference or known also with the name of exclusive disjunction so that the result is one, that is, so that the result is lit

One of them will be on and the other off. One of them, on the other off, and so our results would be on 1. That is the exclusive disjunction that is read. We said a moment ago. How? Symmetric difference. Therefore, if we have an off and an on our result would be on. Here also on. There not here yes.

It is not fixed this "on", this "off" result. "On" and there are no more cases like this. It would be "off". "Off". "Off". "Off". "Off". All this column is the result of that first parenthesis. And we stop focusing on this. We go to this other side. We also have a parenthesis. We would have to find the value of that parenthesis.

With the "y" if both are on the result would also be on. The same as here but if both are not on. The result would be off. Here also off. Here both are off our result is off. It gives us on when both are on. There is a negation in front of this parenthesis if you notice it.

That negation will make it change its values. From this column, let's change it to "0", that is, "off/off". "On/off/on". "On/off/on". Yes, this column would have to connect with this one. But look, we hadn't put the values ​​"r" here. "On/off/on". "On/off/on".

Apagado, encendido, apagado, encendido, apagado, encendido. Nos concentramos en estas dos columnas. Vamos a ponerle de un color. Estas dos columnas que están de color celeste. Con la O. Suficiente que uno de los dos esté encendido. Para que el resultado también sea encendido. Miren. Resultado encendido. Aquí también el resultado encendido. Lo mismo aquí. Aquí también.

Lo mismo aquí. Y finalmente, pero cuando los dos están apagados con la O, el resultado también es apagado. Y bueno ahí tenemos el resultado de lo que estaba dentro de los corchetes. Esto ya no sería necesario. Podemos quitarle el color. Y ahora so solamente. Tendríamos que utilizar las columnas que están de color amarillo. Con la doble implicación que se lee, sí y solo sí.

El resultado estaría encendido. Si los dos son iguales. Si los dos están apagados el resultado es encendido. Aquí también los dos son iguales. Resultado encendido. En este caso no son iguales. Resultado apagado. Aquí también el resultado es apagado. Lo mismo aquí. Bueno en este caso. Es decir en esta fila. Los dos son iguales.

the result would be on. Here they are different, the result would be off. Here in this row, both are on. The result would also be on. And the last row, they are not the same. Therefore, the result would be off. And well, here we have different values ​​between zeros and ones, between on and off. Which means that this

or this column gives as a result a contingency. What we had written down here, tautology is all 1, contradiction is all 0. Contingency mixed its values. And here we see that our final result that we are going to mark there in green color. And to this we will remove its color. As everything is intermixed, this would be a contingency.

And let's go with the following. Exercise 3. In this propositional formula, that is, this compound proposition, we have three simple propositions. P, Q, R. Then we would have eight rows. First, second, third, fourth, five, six, seven, eight. There are eight rows. The first proposition would have four values turned on and four values.

The second proposition would have two values turned on, two values turned off. Again, two values turned on, two values turned off. And the last one turned off, turned off, turned off, turned off, and with this we are going to start assigning its values to each proposition that is within the propositional formula.

Empecemos. Pero tiene cuatro encendidos y cuatro apagados. Pero tiene una negación por delante. Si lo logran ver tiene la negación. Entonces cambiaría a cuatro apagados. Después cuatro encendidos. Buscamos otra P. Está justo aquí. Tendría cuatro encendidos. Cuatro apagados. Ya no hay más P. Pasaríamos a la siguiente que es Q. Dos encendidos, dos apagados.

Dos encendidos, dos apagados. Buscamos otra Q. Aquí está. Dos encendidos, dos apagados, dos encendidos, dos apagados. Pero tiene una negación por delante. Eso cambiaría sus valores. Sería dos apagados. Dos encendidos, dos apagados, dos encendidos. Ya no hay más que pasaríamos a la R. Aquí está la R.

Encendido apagado. Encendido apagado. Encendido apagado, encendido y apagado. Pero tiene una negación por delante, eso cambiaría. Apagado encendido, apagado encendido. Apagado encendido, apagado encendido. Buscamos otra R, aquí está. Encendido apagado, encendido apagado, encendido apagado. Encendido apagado.

Well, once you have assigned the values to each simple proposition, we have to start solving it. First with the parentheses, then with the brackets and at the end keys if there were keys as we have seen in the first exercise. Well, let's start with the first parenthesis. We work this column with this other and they are connected with the "i".

With the "i" both would have to be turned on so that the result is also turned on. Look here this is one, this too. The result one. The same here. The rest would be off. Off, off, off, off. We leave the parentheses. Taking into account that this is the resulting column. Of the parentheses. That we have painted yellow.

This is no longer important, we can remove the color. The same here. And we're just going to analyze or use this column. That we have marked here. As that is the result of what is inside the parentheses. We would have to connect it with this other column. Denegation of R. We are going to change the color of that column a little. There it is.

Estas dos columnas se están conectando con este símbolo que es entonces. Antecedente. Apagado consecuente. Apagado resultado. Si dudas de tus resultados para eso tienes la tabla. De verdad. Esta tabla también se las pusimos en la tarea número uno. Sí. Bueno entonces. Estábamos verificando no en qué casos nos daría. Verdadero cuando tenemos.

the conditional also called implication, that is, what is read then. With this table we would look for this case: antecedent turned on, consequent turned off, result: turned off in the implication, antecedent: 1 consequent 0 in other words, antecedent 1 consequent 0 we look here is 1, turned on then, turned off, result turned off.

Here it would be 1 the same here 1 off then off it would also be 1 1 the same here each one is fixed to write down their result on then off result off on then on result on and here off then off doubts about your result there you can see the table

Here we have these values turned off and with the table of truth, if both are turned off, our result is on. Result turned on. The same here. That would be the result of what is inside the brackets. This. Instead, we can remove the color. There it is. Well, we go to this side to the right.

and we start within the parentheses. We have the "o". We would have to work with this column and this other one because it is not "q" with the "o" enough that one of them is on so that the result is also on. There we have "off", "on", "on". There is a negation in front of that parenthesis. That means that it would change the value of this column.

We will put a different color to differentiate it. We were saying that we focus on this column and as it has a negation ahead, it would change values. It would be off, off, off, off, on, on, off, off. Now we work with the column of R and this column of negation. They are connected with the I.

Para que de, resultado encendido. Los dos tendrían que estar encendidos con la I. Encendido, y no hay más casos como ese. Todo lo demás apagado. Solo había este caso. Bueno, esto de aquí. Sería el resultado final del corchete. Le pondremos aquí un color que resalte. El resto ya no sería necesario. Le vamos a quitar el color. Ahora la columna que está a la izquierda.

With the one on the right, these that are colored. Well, I'm also going to paint this one yellow. And we have the bi-conditional. The result would be here, with the double implication. The two would have to be the same. And we don't have that case. If the two are the same, the result would be lit. But we don't have two the same.

Therefore, this would be 000, that is, off. And voila, this would be the final result. Everything is zero. If everything is zero, it would be a contradiction. And well, in this way you would have to solve your task 1. It may be that you have learned the previous semester with true or false. But in this course,

tendrían que ir adaptándose a lo que se está explicando aquí. Si existiese alguna corrección, algún error o algo que no se haya entendido, deben llenar un formulario de asistencia. En ese formulario de asistencia ustedes escribirán sus comentarios. Ese formulario les digo que es obligatorio. Con eso se va a controlar si por lo menos han atendido la clase.

Simplemente, les faltaría completar los ejercicios de la tarea 1 si existiera algún error en la resolución de estos tres ejercicios y tienen una prueba que en su cuaderno lo hayan hecho y que exista algún error. En esta explicación, pueden ganar puntos adicionales dando su comentario en el formulario de asistencia. Ese formulario se va a publicar en Classroom.

Sigan haciendo los otros ejercicios si aún no lo han hecho. Posteriormente, se les va a enviar otra tarea. Estas tareas solo son para que estudien. El examen va a ser presencial. Algún detalle adicional que existiera para el examen se les va a comunicar en estas clases asincrónicas. Por eso es muy importante que presten atención.

Bien con eso terminamos y tendríamos la siguiente clase asincrónica mañana.