Kenneth Rosen Edition 7 Exercise 1.5 Question 27 (Page No. 66)

Question

Determine the truth value of each of these statements if the domain for all variables consists of all integers.

1. $\forall n \exists m (n^2 <m)$
2. $\exists n \forall m (n <m^2)$
3. $\forall n \exists m(n+m=0)$
4. $\exists n \forall m (nm=m)$
5. $\exists n \exists m (n^2+ m^2 = 5)$
6. $\exists n \exists m (n^2+m^2 =6)$
7. $\exists n \exists m (n+m = 4 \wedge n-m =1)$
8. $\exists n \exists m (n+m = 4 \wedge n-m =2)$
9. $\forall n \forall m \exists p(p= (m+n)/2)$

 

Comments

1 True.(For every integer square we can find a larger integer)
2.True.
3.True.(Since the domain is integer , for n=-m the condition holds)
4.True.(For n=1 , multiplying it with any integer gives the integer itself.)
5.True(4+1=5)
6.False (No such pair exists)
7.False , since n+m=4 . 4 is even , thus n and m both should be even or both should be odd , even - even = even and odd-odd=even . Thus n-m=1 is not possible.
8.True , 3+1=4 and 3-1=2
9.False .

---

how option b is false?

---

My mistake , it'll be true for negative numbers.