# Determine truth value of following,

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

• Why don't you start us off with an idea you have about the four separate questions. Most of us would be happy to confirm, or correct any misstep you might make. But you need to be as active as any helper is with the aim you are able to answer these questions. – amWhy Nov 29 '16 at 20:44
• 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 @amWhy – Somesh Pursnani Nov 29 '16 at 20:50
• What big set does x belong to, in general? – amWhy Nov 29 '16 at 20:59
• sorry, x what ? – Somesh Pursnani Nov 29 '16 at 21:01
• $x$ must be an element in some set like the natural numbers, or at least $x\in \{1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18\}$, from which you can determine whether all are even, or whether their exists an x that is prime... – amWhy Nov 29 '16 at 21:04

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 $(1)\;F\;\; (2)\;T\;\; (3)\;T\;\; (4)\;F\;\; (5)\;F\;\;$