Let A be the set Consisting of the Numbers 1, 2 3, 4 and five. In task one. We want to determine the truth value of each of the following propositions A. X. Six Eggs. in a search that eggs for three equal 10 for bebe for all X. In a X plus three is less than 10 parsi exists X. In a Such that expert three is less than five. And party for all eggs In a explore three is less than or equal to seven. And we want to find a counter example in B. And see if it exists in task two. We negate each proposition in task one. So let's start with subtask one. Mm So yeah there in proposition A exist X in a With the property that explore 3, 7. The equation X plus three equal seven is the same or is equivalent to X equal Sorry, Explosive three Equal 10. It's equivalent to X equal seven because X if X plus three is equal 10 then it's got to be equal seven. That's the only value that has this property here. The only solution to this equation. So uh the proposition in part they will be true if seven is a. Is an element of the said hey but that's not the case. So proposition is false. Mhm Sim. Political seven is not in the set a. and seven is the only value of X For which explores three equals equals 10. So let's see party for all X. In a explode three is less than 10. We see that explodes three. Less than 10 implies immediately that X is less than 10 -3 equals seven. That is X. Less than seven. So the propositions for all X. M. A. Explore three less than 7 6 equivalent to for all eggs in A. X. is less than seven. This proposition here is equivalent to this one Because the inequality extra three less than 10 In place immediately. That eggs is less than seven. Mhm. In fact we can go from this inequality to this other inequality by um Are in three. Both sides of the inequality. Okay so Let's see if all the elements in a are less than seven. That's true. 1234 and five. All of those numbers Are less than seven. So part B is true. Okay so let's go for a party exist X. In a. Such that X plus three is less than five And we do the same explore three. Less than five. In fact here let's put here equivalent in order to know that they are the same here we use again equivalent. That is because we can pass from one equality to the other by using an algebra operation in this case these three we pass it to the right so X is less than five minus three and that's the same as X. Listen to if we start here we write to as five minutes three and at three both sides and we get this. So they are equivalent. So the proposition for all X in a explosive three, sorry, exists X in a such that X plus three is less than five is the same as exist X in a Such that X is less than two. This proposition is equivalent to this one in part C. And that's because this inequality Is equivalent to this one. And let's see if there is an element in A which is less than two. That is true because we have 11 is less than two and because there is we use the operator exists. We know that it's true. That is at least one element with the property. Then the proposition is true. You're talking about part see here, so part a false party is true and part C is too also. So let's go to party for all X. And a extra three is less than or equal to seven. So we're all eggs in a X plus three Less than a report to seven is equivalent to for all X in a X is less than or equal to the past. These three to the right 7 -3 is four. All the elements. So this will be true if all the elements in a are less than or equal to four but that's not true because there is at least one element which is not less than or equal to four. That is five. So this is false because five is in a N five is not Less than or equal to four. That is his greater. 30 is creator than four as we can see this foot here first. Okay, dad, so in summary for AIDS falls per piece true part is true and party is false. Each time we have a false statement we have a country example that is part A for example exists X in a such as X plus three is 10. In this case we know that that's false because all the elements in a do not verify this equality so we can find country examples in the false cases. In this uh case we want to contract sample in B and B. It's true that is we have no counter examples. That is all the elements in a verify this inequality so there is no kind of example But indeed there is a cut for example. And that counter examples we have found here is five. So let's put it here. B Yes, no, sorry. Yeah, in part B there is no counter example because the proposition there yeah is true and the counter example in for d is five, that is that's the element that contradicts the proposition. There is an element which is not but which has now the property that the element plus three is less than or equal to seven. Okay, so let's go now to subtask number two and we're going to negate all the proposition seen stuck and sub task one. That is in part A I'm going to write only negation. So let's see the proposition for A.S. an element in a with its property experts recall 10, that's the negation of that is all the elements in a failed to have that property. That is all the elements in a Are such, that exploratory is not 10. There is an indication of that's right correctly here, this part negation is just in this case, particular case, the construction has quantified operator exists or for all which is changed to the other in the negation and the investigation of the property in this case, simple thing can be more complex for in this case very simple as that and that's logical because it's if uh proposition is true or false, the other has the of the value of the truth value. So for example, here part A is false and It means that the indication of that is true and the negation, all elements in a has a property that extra Serie is not 10. We can verify it's true because if we had three to all of these elements, we never get the value chain. Okay, as an asian of a nation of B is B is for all X in a extra three at less than 10. So the negation is exists an element in a such that export three is not less than 10. It's not usual to write it this way, but in this other equivalent form that it exists an element in a Such that if express Aries is not less than 10, Then X-plus three is greater than or equal to 10. So this maybe it's a better way to write it, parsi negation of exist on an element in a with the property that expose three is less than five. Negation will be all the elements in A has a property that Explore three East greater than or equal to five. Hicks was three. It's not less than five as I wrote it before. And then I put the equivalent forum which is more is extra three. Sorry, X plus three. It's not less than five. It is created than or equal to five. No party. Yeah, for extra extra sarees less than or equal to 77 delegation is existent X in a such that X-plus three is not less than or equal to seven. That is exists an X in a such debt, Explore three is greater than seven and this proposition is true because this is false and this is true. That is there is an element. We can find an element in a with the property that that element plus three is greater than seven. That element is five because five Plus three is 8 which is greater than seven. Okay, that's it. We have delegation vault. He propositions and we have discussed about the truth value of fish of the given propositions in parts A B C and D. For a fools. Part B is true, part C is true, and parties falls. The counter example in party is five, And here are the negations of each of the propositions in part a 30.