this is the question i'm working on:
Given that U1, U2, U3 are domains for m, n and o respectively, determine weather the following statements are true, false, or not propostion.
U1: {2,6}
U2:{-2,3,10}
U3:{5}
l(m,n): m ≤ n
s(m,n,o): m+n=o
e(m): m is even
the statements:
A)∀m ∃n ( e(m) ∧ l(m,n) )
B)∀m ∀n ∃o s(m,n,o)
C)∀o ∃m ∃n s(m,n,o)
my answers were:
A)not proposition
B)False
C)True
but I'm still not sure could someone help me?
$A$ is $True$; because all elements of $U_1$ are even and for every element in $U_1$ there exists at least one element in $U_2$ that's larger or equal to the corresponding element in $U_1$.
$B$ is $False$; because we need to find that for every element in $U_1$ and $ U_2$ we can take combinations of elements and sum them to $5$ (elements in $U_3$). A counterexample to prove that it's false, is that $6$ from $U_1$ can't be summed with $-2$ or $3$ or $10$ to get $5.$
$C$ is $True$; because we need to find one element from $U_1$ and $U_2$ each that sums to the elements in $U_3$, which is $5$. So we have $2 + 3 = 5$, which works.