I have to decide whether the following statements are true by using a proof or a counter-example.
$$\begin{align} &(\text{i})~~~ a < b \text{ and } c < d \Rightarrow ~a - c < b - d\\ &(\text{ii})~~ a < b \text{ and } c < d \Rightarrow ~ac < bd\\ &(\text{iii})~ a < b \text{ and } c < d \Rightarrow ~ac^2<bd^2 \end{align}$$
I've got following solutions. While I am sure that they are enough, I think that these ones could be "formally cleaner" Both here and on the paper.:
$$\begin{align} &(\text{i})~~~ a = 1~ b=2~c=-5~d=1 && 1+5 = 6 \not\lt -5-1 = -6 \Rightarrow Statement ~(i)~ is ~false. \\ &(\text{ii})~~ a = 1~ b=2~c=-5~d=-10 && 1*(-5) = -5 \not\lt -2*(-10) = -20 \Rightarrow Statement ~(ii)~ is ~false.\\ &(\text{iii})~ a = 1~ b=2~c=-5~d=1 && 1*5^2 = 6 \not\lt 2*1^2 = 2 \Rightarrow Statement ~(iii)~ is ~false. \end{align}$$
The first one: $1<2\wedge -2<-1$, but $1-(-2)=3 = 2-(-1)$.
The second one: $-2<1\wedge -2<1$, but $(-2)(-2)=4>1 =1\cdot 1$.
The third one: $1<2\wedge -3<-1$, but $1(-3)^2 = 9 > 2 = 2(-1)^2$.