Feedback on hw 2

2.4

• A lot of people are confused about how to negate an implication. From a truth table, one can see directly that the negation of "P implies Q" is "P and (not Q)". Also using a truth table, one sees that the negation of "P implies Q" is not "P implies (not Q)", nor "(not P) implies (not Q)" .
• Also, please make sure that whatever you write makes sense in plain English. For example, " There is x in A for all b in B such that b ≤ x " does not mean anything. One way to write it would be " There is x in A such that for all b in B b ≤ x ".

2.10

• There was again a lot of confusion on the negation of an implication. Here is a reminder:

  • The converse of "P implies Q" is defined to be "Q implies P" . It is not equivalent to "P implies Q", but it is not its negation either.
  • The contrapositive of "P implies Q" is defined to be "not Q implies not P" . It is equivalent to "P implies Q", so it cannot be its negation.
  • As noted above, the negation of "P implies Q" turns out to be "P and not Q".

  Each of the three statements above are different.

2.24

• A few students forgot to argue why (a) implies (b) . Whenever you make a claim, do not forget to justify it !

2.52

• From x ∈ (A ∩ B) - (A ∩ C) ⊆ (A ∩ (B - C)), a lot of students claimed that x ∉ C. This is true, but this must be justified: the only immediately clear statement one can make is that x ∉ A ∩ C, which is in general weaker than saying x ∉ C.
• Some students mixed up set and logic notation: recall that even though they behave similarly, ∩ and ∧ mean different thing: one is an operator on sets, the other is an operator on true/false statements.