In class, we were introduced to the topic of transitivity. This idea simply means that if a ⇒ b, and b ⇒ c, then a ⇒ c. As simple as this idea may sound, the professor asked the class to negate the statement:
and simplify it by transforming all implications into disjunctions as a way of proving the entire original implication to be true. Personally, this is the hardest part of Week 4’s lessons since it involves De Morgan’s Law and many other laws we learned in the previous week, such as the distributive laws:
The other difficulty in simplifying the negation is that the notations become really messy as the implication gradually breaks down into disjunctions. Hence, it was very hard to not make a mistake. In my picture example, I negated the implication by changing the P --> Q to P or not Q following De Morgan’s Law. On the third line, I changed P ---> Q into its equivalent statement not P or Q. I also simplified the negation of the last bracket by employing De Morgan's Law. On the fifth line, I expanded the equation using a similar approach to FOIL in math class. Since all statements are conjoined by or's, Q or not Q is omitted from the equation since it is false. (recap: false or true yields true, however false and true yields false.) On the last line, since all three statements are false, the negation of the original statement is true. Hence, the original implication is true.
To sum up, the most challenging part about this week's class is to simplify or change the original statement using De Morgans law, distributive law and etc. One mistake during computing could lead to a completely different conclusion. This reminds me that I have to be more careful during future tests.
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