# Thread: Truth value of the statement

1. ## Truth value of the statement

P(x,y) means "x + 2y = xy", where x and y are integers.
Determine the truth value of the statement.

(a) P(1,-1).
(b) P(0, 0).
(c) ∃y P(3, y).
(d) ∀x ∃y P(x, y).
(e) ∃x ∀y P(x, y).
(f) ∀y ∃x P(x, y).
(g) ∃y ∀x P(x, y).
(h) CS173 Discrete Mathematical Structures ¬CS173 Discrete Mathematical Structures ∀x CS173 Discrete Mathematical Structures ∃y CS173 Discrete Mathematical Structures ¬P(x, y).

2. Hello, captainjapan!

I'll have to make some guesses . . .

$\displaystyle P(x,y)$ means "$\displaystyle x + 2y \:=\: xy$", where $\displaystyle x$ and $\displaystyle y$ are integers.

Determine the truth value of the statements.

$\displaystyle (a)\;P(1,-1)$
We have: .$\displaystyle (1) + 2(-1) \:=\:(1)(-1)$ . . . True

$\displaystyle (b)\;P(0, 0)$
We have: .$\displaystyle (0) + 2(0) \:=\:(0)(0)$ . . . True

$\displaystyle (c)\;\exists y\: P(3, y)$
We have:
. . There exists $\displaystyle y$ such that: .$\displaystyle (3) + 2(y) \:=\:(3)(y)$ . . . True

$\displaystyle (d)\;\forall x\:\forall y\:P(x, y)$
This says: $\displaystyle x + 2y \:=\:xy$ for all $\displaystyle x$ and $\displaystyle y$ . . . not true.

$\displaystyle (e)\;\exists x\:\exists y\:P(x, y)$
There is an $\displaystyle x$ and $\displaystyle y$ such that: $\displaystyle x + 2y \:=\:xy$ . . . True
[See parts (a) and (b).]

$\displaystyle (f)\;\forall y\:\exists x\: P(x, y)$
This says: for any $\displaystyle y$, there is an integer $\displaystyle x$ such that $\displaystyle x + 2y \:=\:xy$ . . . not true.

If $\displaystyle y = 4$, we have: .$\displaystyle x + 8 \:=\:4x\quad\Rightarrow\quad x = \frac{8}{3}$ ... not an integer.

$\displaystyle (g)\;\exists y\:\forall x\:P(x, y)$

This says: for any $\displaystyle x$, there is an integer $\displaystyle y$ such that $\displaystyle x + 2y \:=\:xy$ . . . not true.

If $\displaystyle x = 5$, we have: .$\displaystyle 5 + 2y \:= \;5y\quad\Rightarrow\quad y \:=\:\frac{5}{4}$ ... not an integer.