I want to represent the statement "Some numbers are not real " using quantifiers. I have been told by my teacher that the correct way to represent this is
num(x) : x is a number
real(x) : x is real
∃x (num(x) ^ ¬real(x))
This made sense, i can translate this statement into "There exist some x such that x is a number and x is not real.
But the answer i came up by myself is this one
∃x (num(x)=> ¬real(x))
In translation , There exist some x such that if x is a number then x is not real.
I just can't get around why my answer is wrong, for some x ; if x is a number then x is not real. Doesn't that sound very similar to the statement "Some numbers are not real".
In one of the video lectures i saw this example which made me even more confused.
"No dog is intelligent"
dog(x) : x is a dog
intel(x) : x is intelligent
The representation was
∀x (dog(x) ==> ¬intel(x))
if this representation is true, how is my representation of "Some numbers are not real" wrong.
PS : I am just a beginner at Discrete math finding my way, please pardon me if the question doesn't meet the quality standards of the community.