Any help would be appreciated: Quantifiers and Domains (Discrete Mathematics) Fo

Question

Any help would be appreciated: Quantifiers and Domains (Discrete Mathematics)

For each of the following state whether or the propositions are true or false for the specified domains.

Reminder:

N is the natural numbers (positive whole numbers including zero)

Q is the rational numbers (numbers that can be written as the quotient of two whole numbers)

R is the real numbers (includes irrational numbers such as 2).

(1) xy : x < y

(a) (2 point) x, y N (same as x N, y N)

(b) (2 point) x, y Q

(c) (2 point) x, y R

(2) xy : x < y

(a) (2 point) x, y N

(b) (2 point) x, y Q

(c) (2 point) x, y R

Explanation / Answer

xy : -- this means that for ALL x , there exists Y such that x < y

(a)  x, y N

xy : x < y this can be taken in this order

i) x set to any natural number

ii) AFTER x is set we can find atleast one Y such that x < y

basically, this means that if we get any natural number x we can always find a y which is greater than x! this is TRUE since natural number are from 1 to inifinity , if we select 1 as we can get 2 for y and so on.. there is no upeer bound for natural numbers so , this will always hold.

(b)  x, y Q

this means that we can get any rational number y such that x < y after we have selected x FIRST.

since raational numbers are unbounded from -infinity to +infinity , any x that you may pick , we can always find a "y" which will be bigger . SO TRUE.

(c)  x, y R

same as rational numbers . if i pick 2 = 1.414 as x i can pick 3 = 1.73 as y.

so, it'll also hold , so TRUE.

-------------------------------------------------

xy : x < y   

(a) x, y N

here the ordering has changed, let us look at it

i) ATLEAST 1 x can be found before , and this would have to work for all possible y's.

ii) after this x is SET, we can put any y, andd it must satisfy x < y.

this is FALSE. because if we set x as 5 , y for all natural numbers (include 1 2 3 and 4) must satisfy x < y;

(b), (c) will be FALSE similaryly.

for Q , set x as 1/2 = 0.5 , then y which ccan even be -1/2 = -0.5 will not satisy x < y.

for R, x as -1.414 , then y as -1.73 will not satiisfy.

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thank you

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