I am in math and statistics and have taken everything from linear algebra to multi variable calculus and just started a mathematical reasoning which is really making my head spin (in a good way) but I am second guessing myself quite a bit.
Question:
Let $U$ be the universe under sconsideration, and let $P(x)$ and Q(x) be predicates with free variable $x$. Find the negation of each statement.
a. $$(\forall x\in U)(P(x) \Rightarrow Q(x))$$
answer: $$(\exists x\in U)(P(x) \bigwedge \sim Q(x))$$
b. $$(\forall x\in U)(Q(x) \bigvee P(x))$$
answer: $$(\exists x\in U)(\sim Q(x) \bigwedge \sim P(x))$$
c. $$(\exists x\in U)(Q(x) \bigwedge P(x))$$
answer: $$(\forall x\in U)(\sim Q(x) \bigvee \sim P(x))$$
d. $$(\exists x\in U)(Q(x) \bigwedge P(x))$$ (for d. use an implication in your answer)
answer: $$(\forall x\in U)(\sim Q(x) \bigvee \sim P(x))$$
I am pretty sure i got all of them right except for d. as I am not sure what it means to use an implication in my answer.