Question

Consider the statemen Pr0,0 such that ya Q: y > 0 such that Vz, > 0, y < x S: 3yER such that Vx R,V: ER.y»>a Select all true statements Consider the following formal "proofs" and select all those which are correct. Proof that Ris te. Let r e R be arbitrary and taker - 1. Then if , we have:>x. This proves R Proof that R is ue Let r & R be arbitrary and take y-1. Then if y, we have>a. This provesR Proof that R is false. Take 0 ad y2. Then taking we have : > y and : < r. This R is false. Proof that R is false, Take ad leteR be arbitrary, Then there exists : E R with 0 < : < y so that Rİs false. Proof that S is tre. Let y eR be arbitrary and takery -1. Then if :>t, we have : > a. This proves S Proof that S is false. Let ye R be arbitrary. Take y2 audy +1 so that>y and S When looking at more compllcated statements it often helps to break them up into pleces we can understand, for example conslder the last part of statement R. For fixed values of t, R select all of the following statements equivalent to

Answer 1

(The picture is not clear ,but I have tried my best to understand the questions properly and answer it.)

In the first question to see whether a statement is true or false we try to prove a true statement and try to give a counterexample if a statement is false.

In the second question we try to find flaws or wrong argument in the proofs. If there is some such wrong argument then the proof is Incorrect, Otherwise correct.

In the last question to prove equivalence we have to prove that the conditions are if and only if, that is if any of the two statements is assumed then the other statement can be proved.

In the first question to see whether a statement is true or false we try to prove a true statement and try to give a counterexample if a statement is false. In the second question we try to find flaws or wrong argument in the proofs. If there is some such wrong argument then the proof is Incorrect, Otherwise correct. In the last question to prove equivalence we have to prove that the conditions are if and only if, that is if any of the two statements is assumed then the other statement can be proved. \r\n P: Forall x > 0, exist y > 0 such that y < x True: Let x > 0 Then x/2 > 0/2 = 0 [Since 2 > 0] Choose y = x/2 (> 0) Clearly x > y > 0 So, P is true exist y > 0, such that Forall x > 0, y < x False: Let if possible Q is true, i.e., exist such y(> 0) Choose, x = y/2 Then x > 0, but x < y