The Week in Review: What is negation?

This week, we began learning more symbols and terminology.

On Monday's lecture, I was stuck on standard negation. It was Monday morning and I was feeling a bit tired. In class, I looked at the corresponding slide and tried to understand the concept, but to no effect whatsoever. My class was told to try and negate the following statement in two minutes:

¬(∀ x ∈ X, ∃ y ∈ Y, P(x) => Q(y))

I was out of it. I thought, I can't do this in two minutes! I need more than that! I copied the statement down in my notebook and tried to think about it. Eventually, time was up. This is the solution:

∃ x ∈ X, ∀ y ∈ Y, P(x) ∧ ¬ Q(y)

I wrote down the solution below the original statement, but I was still stuck. I talked to Professor Heap after class. As he explained, I realized I was overthinking. It was all review on negation. I then felt relieved, knowing that it wasn't as hard as I thought it would be. I regained my confidence.

Negating a statement means to take the opposite of the original statement. The negation (or opposite) of a universal claim (For all...) is an existential claim (There exists...) and vice versa. As well, the only time a statement is false is when the antecedent, P(x), is true, but the consequent, Q(y), is false. In this case, the original statement says that P(x) implies Q(y). However, to negate this, Q(y) has to be false while P(x) remains true in order to make the statement false because that's the only situation when this happens. (For all other cases, the statement is true.) Thus, we would have to put the "¬" symbol in front of Q(y). 

The next morning, I had my second tutorial. Our exercise involved writing English statements using logical symbols as well as determining true and false statements with explanations. The concept of math logic was still quite new to me, but I did make sure to ask my TA questions on things I found difficult. Once again, I hope I did well on my quiz, given that it was similar to the first question on the exercise. I did try my best.

I do find the idea of switching statements between math and English quite fascinating. It's something I have never seen before. Initially, I was lost when it came to this idea, but after getting proper help from Professor Heap and the TA over the weeks, I seem to be enjoying it now.