# What do these statements mean in discrete mathematics?

Example 1: Let P: Z x Z → {T,F} where P(x,y) denotes "x+y=5"

Example 2:

Let B = {T,F}. Let P(p,q,r,...) be a proposition. Then, P := (p → q) → r: B x B x B → B

• What notation, exactly, are you confused about? There are at least 10 different symbols and notions here. The source of these examples likely has their definitions or a glossary of symbols. – user296602 Oct 20 '18 at 23:19

## 2 Answers

Example 1: The set of ordered pairs of integers such that the sum of the first and the second is 5. P(x,y)=xPy which means P is a 2-place predicate defined as x+y=5.

Example 2: The second one is the Boolean definition of Predicate. It, in essence, says given a relationship P such that p implies q whole implies r will give you an ordered triple such that the whole will be an element B={T,F}. That is, the Predicate P will be either true or false given the values of (p,q,r…).

This is what I understood of it.

Example 1: is a very abstract and rather obscure but necessary way to say:

"$$x + y =5$$ is either true or false"

"Let $$P...:$$" means let $$P$$ be a way of writing a statement.

"Let $$P:\mathbb Z\times \mathbb Z \to...."$$ means let $$P$$ be a way of writing a statement about two integers. That is to say the input of $$P$$ will be $$(x,y)$$ a pair of integers. So $$P(x,y)$$ will be a statement about $$x$$ and $$y$$.

"Let $$P:\mathbb Z \times \mathbb Z \to \{T,F\}$$ means the output of the statement will be either TRUE or FALSE. So given $$x$$ and $$y$$ the the statement $$P(x,y)$$ will either... be TRUE or FALSE.

So sayings $$P(x,y)$$ means "$$x + y = 5$$" Means that we will evaluate any $$P(x,y)$$ as "Does $$x + y = 5$$ and we will determine that either ... yes, $$x + y = 5$$ or ... no, $$x + y$$ doesn't $$=5$$>

For example $$P(2,3) = T$$ because $$2+3 = 5$$. And $$P(4,2) = F$$ because $$4 + 2 \ne 5$$.

....

The idea is that we are trying to use the algebraic concept of function in a logical context.

In this case "{{A}}+{{B}} = 5" is a statement that takes to integers as input and outputs a single T/F value. In terms of functions this means $$P$$ is a function that maps an ordered pair of integers to a single T/F. So the domain of $$P$$ is $$\mathbb Z\times \mathbb Z$$ and the codomain of $$P$$ is $$\{T,F\}$$.

Example 2:

We have three statements $$p, q,$$ or $$r$$ and each of those are either $$TRUE$$ or $$FALSE$$ and we want to know if the statement $$(p\to q)\to r$$ is $$TRUE$$ or $$FALSE$$.

So $$P$$ is the the statement $$(p\to q) \to r$$.

The input of $$P$$ is $$(p,q,r)$$ (not $$(p,q,r,....)$$ by the way; you typed that wrong) which is an ordered triple of TRUE/FALSE values. And the output is a single output TRUE/FALSE value.

So $$P$$ is a function. If $$B$$ is the set $$\{T, F\}$$ then the domain of $$P$$ is $$B\times B\times B$$. That is the set of all possible triplets. That is $$B\times B\times B = \{(T,T,T), (T,T,F), (T,F,T), (T,F,F), (F,T,T), (F,T,F), (T,F,T),(F,F,F)\}$$. The codomain of $$P$$ is $$B = \{T,f\}$$.

And the function is defined as $$P(p,q, r)$$ as $$(p\to q)\to r$$.

For example $$P("bats\ are\ birds", "pigs\ eat\ corn", "Dogs\ teach\ algebra") = P(F, T,F) = (F\to T) \to F = T \to F = F$$.

If we wanted foralize:

$$P$$ is the function:

$$(T,T,T) \mapsto T$$

$$(T,T,F) \mapsto F$$

$$(T,F,T) \mapsto T$$

$$(T,F,F) \mapsto T$$

$$(F,T,T) \mapsto T$$

$$(F,T,F) \mapsto F$$

$$(F,F,T) \mapsto T$$

$$(F,F,F) \mapsto F$$

[Math does get fairly abstract. But it always makes sense.]