Question

Suppose P (x, y) means “ x and y are real numbers such that x + 2y = 5 .” Determine whether the statement is true for ∀x∃yP(x,y) and ∃x∀yP(x,y)

Answer 1

Suppose P (x, y) means “ x and y are real numbers such that x + 2y = 5 .”
Determine whether the statement is true for ∀x∃yP(x,y) and ∃x∀yP(x,y)

∀x∃yP(x,y) - false
Above statement implies that all values of x, there exists some value y, such that x + 2y = 5
But actually it is not possible to find a value for some specific values of y.
i.e. there exists some value x, for all values of y for which x + 2y != 5

using example :
for, x = 0 there is no y s.t. x + 2y = 5
for, x = 2 there is no y s.t. x + 2y = 5
for, x = 4 there is no y s.t. x + 2y = 5
for, x = 6 there is no y s.t. x + 2y = 5
for, x = 8 there is no y s.t. x + 2y = 5
...
for x = 100 there is no y s.t. 2y = 5
and so on

so for all the even values of x there is no y s.t. x + 2y = 5
So∀x∃yP(x,y) - false for even values of x where x and y belongs to real number.

∃x∀yP(x,y) - true

Above statement implies that for there exists some x, for all values of y, such that x + 2y = 5

using example :
for, y = 0, x = 5
for, y = 1, x = 3
for, y = 2, x = 1
for, y = 3, x = -1
for, y = 4, x = -3
...

for x = 10, y = -15
and so on

So, ∃x∀yP(x,y) - true where x and y belongs to real number.