Determine truth value of following:
$(1)\;\forall x P(x)$
$(2)\;\exists x Q(x)$
$(3)\;\forall x\, \exists y\;R(x,y)$
$(4)\;\exists x \,\forall y\;R(x, y)$
$(5)\;\forall x\,(\lnot Q(x))$
For $x, y \in \mathbb Z^+$, (meaning $x, y$ are positive integers):
Let $P(x): x$ is even; $\quad\;Q(x): x$ is a prime number; $\quad \;R(x, y): x+y$ is even.
My Understanding: p(x) = 2,4,8,10 q(x) = 3,7,11,13,17 not sure on r(x) Ans: i. false as x is all postive integers and all are not even ii. true. atleast one x which is prime iii. iv. v.False. x are set of postive integers, negation of p(x) is odd integers
Let $x, y \in \mathbb N$.
Let $P(x)$ denote "$x$ is even."
Let $Q(x)$ denote "$x$ is prime."
Let $R(x, y)$ denote "$x+y$ is even".
Now, clearly, we know that not all integers are even. Hence $\lnot\forall x (P(x))$. That is $\forall x (P(x))$ is false.
We know that there are many prime numbers in the integers. Example: For $x=7\in \mathbb Z, Q(7)$ is true. That means that $\exists x Q(x)$ is true.
(3) It is true that $$\forall x \exists y (R(x, y)).$$ for every integer $x$, there is some integer $y$ such that $x+y$ is positive. For all even $x$ choose y to be any even number and we have $R(x, y)$ is true. Similarly, for all odd $x$, there is some odd y, so that $x+y$ is odd + odd = even, and therefore true.
The next two questions evaluate to false.
$\exists x \forall y R(x, y)$ is false.
$\forall x(\lnot Q(x)) \equiv \lnot \exists x(Q(x))$ is false
In summary $(1)\;F\;\; (2)\;T\;\; (3)\;T\;\; (4)\;F\;\; (5)\;F\;\;$