# How can I express this statement with mathematical symbols?

I'm trying to express the following statement using mathematical symbols:

The base time in seconds is the block's hardness multiplied by 1.5 if the player can harvest the block with the current tool, or 5 if the player cannot.

If I represent the base time as $$B$$, the hardness as $$h$$ and the optimal tool as $$t^*$$ then I can start the statement as:

$$B = h \cdot ()$$

However, that's where I get stuck. From a code standpoint, it's easy to express as:

if (t*)
b = h * 1.5
else
b = h * 5


I thought using the symbol for implication (I think) would work, but after further research, it reads as true or false, not conditional assignment. For example:

$$x = 2 \implies x^2 = 4$$

This statement evaluates to true because it is read as (I think):

$$x$$ is equal to $$2$$ if $$x^2$$ is equal to $$4$$.

Meanwhile $$x^2 = 4 \implies x = 2$$ evaluates to false because $$x$$ could be $$-2$$.

How can I translate this statement using mathematical symbols only?

Note: This is for recreational practice.

• No. $x = 2 \implies x^2 = 4$ might be read as "$x^2 =4$ if $x=2$" or "$x =2$ only if $x^2=4$" Sep 29, 2021 at 16:01
• $T\to B=h\cdot5$ and $\lnot T\to B=h\cdot 1.5$. I understand this is just for your recreational understanding, but this isn't what mathematics is about, in my opinion. Sep 29, 2021 at 16:01
• @DonThousand can you write that up in an answer for future readers and explain how and why that works? If not I can do the research for each symbol and try to do it justice. Sep 29, 2021 at 16:06
• @Tacoタコス My point is, a future user should never learn from this thread, as this is not how one should do mathematics. Sep 29, 2021 at 16:49

I saw the comments on the origional question, and thought I would elaborate on one of the answers

$$T$$ is a boolean, and is either true or false. If $$T$$ is true, then $$B = h\cdot 1.5$$. This give us

$$T \rightarrow B=h\cdot 1.5$$

Or in other words, $$T$$ being true implies $$B = h\cdot 1.5$$. If $$T$$ is false, then $$\lnot T$$ is true, and $$B = h\cdot 5$$. This gives us

$$\lnot T \rightarrow B=h\cdot 5$$

Or in other words, $$T$$ being false implies $$\lnot T$$ is true, which imples that $$B = h\cdot 5$$.

Another way to do this would be to use the notation for piecewise functions; if we consider $$B$$ to be a function of the current tool $$t$$, and the optimal tool to be $$t^*$$, then we can write: $$B(t)= \begin{cases} 1.5h, & t=t^* \\ 5h, & t\neq t^* \\ \end{cases} \$$

• I like this from the perspective of brevity! Sep 29, 2021 at 16:34