 Hi, and how are you all today? My name is Priyanka and I shall be helping you with the following question. Here, the question says check whether the following pair of statements are negation of each other. Give reason for your answer. Now, here the first statement which is given to us is x plus y is equal to y plus x is true for every real number x and y and its negation given to us in the question says there exists real number x and y for which x plus y is equal to y plus x. Now, to solve this question, we shall first write the negation of the first statement by using the phrases like does not exist or it is false or just writing not or does not. According to the statement and then we will compare it with the other statement to decide whether they are negation of each other or not understood. So, the first step to write down first of all the statement. So, the first statement which is given to us is x plus y is equal to y plus x is true for every real number x and y. Right? Now, the negation of the statement this y is not equal to y plus x for real number for real numbers y or we can say that there exists a real number or we can say that there exists real numbers and y for which x plus y is not equal to y plus x. Now, whereas the statement second let me write down is given to us as there exists real numbers and y for which x plus y is equal to y plus x which is different from the negation which we have formed in part one. Hence, we can see that the given pair of statements are not negation of each other. Finally, we can write our answer as no. The negation of statement in first is, in first is there exists real numbers x and y for which x plus y is not equal to y plus x instead of the statement given in the second statement. Right? So this completes the answer to the solution. Hope answer to the question. Hope you understood the concept. Well, have a nice day ahead.