Let AB CD and xA Prove that if xnot element ofD then xB a Cr

Let A\\B CD and xA. Prove that if x(not element of)D then xB.

(a) Critique the following student’s proof. Indicate all the errors, as you would grade a homework in details on a 6 point scale. Indicate the mistakes between the lines and also comment on them below. Your comments should consist of complete English sentences. In this part you do not need to fix the proof, only comment on the errors.

Can you please answer in sentences I am having trouble with this

Suggested proof to critique: Suppose x A and x(not an element of)B.

By definition of difference x A\\B. By definition of subset, for all x, if x A\\B then x CD

By definition of intersection for al lx, xC and xD. Thus xD

Solution

Suppose x A and x(not an element of)B. (Comment: we need to prove that x B, so we should not assume x is not in B)

By definition of difference x A\\B.

By definition of subset, for all x, if x A\\B then x CD

By definition of intersection for all x, xC and xD. Thus xD (Comment: It is asked to prove that x B if x not in D and here \"x not in B, then x D\" is not a contradiction of \"if x is not in D then x B\")

Possible proof:

Let x not in D. Then,

By definition of intersection of two sets, x not in CD

Now, Since A\\B CD and x is not in CD, implies, x is not in A\\B

Since, x A and x is not in A\\B,

Thus, by defintion of difference of two sets, we have, x B