Manipulations in boolean algebra

Question

In a normal algebraic equation, we can cancel out stuff from both sides of an equality for example, 1 + x = 1 + y implies x = y. However, this is not true in boolean algebra. Can someone give me an intuition for this?

Thanks in advance!

Answer 1

In boolean algebra $1$ means true and $+$ means "or".

True or (any statement) is always true, hence even though we have

True or $\color{blue}{\text{True}} =$ True or $\color{blue}{\text{False}}$

We can't simply say that $\color{blue}{\text{True}} =\color{blue}{\text{False}}$

It is like even though $0(1)=0(2)$, we can't simply conclude that $1=2$.

For the context of real number, note that $f(x)=1+x$ is an injection, hence if $f(x)=f(y)$ we can conclude that $x=y$ but this is not the case for Boolean algebra, it is a many-to-one map.