This week we first talked about the bi-implication and its negation. Bi-implication can be written as A <=> B. This is equivalent to (A => B) ∧ (B => A) or we can write this as (¬A ∨ B) ∧ ( ¬B ∨ A). The negation for bi-implication is (A ∧ ¬B) ∨ (B ∧ ¬A).
Then we talked about some thing about how to prove a universally-quantified implication is true. If we want to prove A => B, the key to prove this implication is that we need to find the relationship between A and B, in other words, there must be some common traits between A and B. We can use Venn diagram and truth table to help us find the common traits between A and B, and then we can judge which statement is true. For example, if we find that A ⊆ B on Venn diagram, we can say that A => B.
When doing assignment 1, I find it is hard to judge when to use the imply symbol and when to use and. Because according to the truth table, "∧" and "=>" are always the same except vacuous truth. In addition, use vacuous truth is a quicker way to find examples or counter-examples.
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