Question About Math Symbol

Question

I want to know what are those symbol that are marked in red...what are their name and functionality?

Consider the finite set of binary strings

{ (000000),(100000),(110000),(111000),(111100),(111110), (111111),(011111),(001111),(000111),(000011),(000001) }

Explain how it is possible to represent such a set in a propositional formula and find the most compact representation.

Solution.

Language For each 0 ≤ i ≤ 5, bi is a proposition, which intuitively means that the i-th bit has value 1. Obviously, ¬bi means that the i-th bit does not have value 1, and thus it has value 0.

Axioms A possible (compact) representation of the finite set of binary strings is given by the following formula:

Answer 1

You can interpret and name them as the 'logical or' $\lor$ and 'logical and' $\land$ of a range of indexed expressions. Just like $\sum$ is the symbol for a sum.

Answer 2

In terms of logical propositions they are related to $\lor$ (OR) and $\land$ (AND), respectively, as $\sum$ is related to $+$:

$$\bigvee_{k=0}^5a_k=a_0\lor a_1\lor a_2\lor a_3\lor a_4\lor a_5\;,$$

and

$$\bigwedge_{k=0}^5a_k=a_0\land a_1\land a_2\land a_3\land a_4\land a_5\;.$$

In terms of bit strings, however, they are simply bitwise $\sum$ (sum modulo $2$) and $\prod$ (product). If $a_1=01010$, $a_2=11011$, and $a_3=00011$, then

$$\bigvee_{k=1}^3a_k=01010\oplus 11011\oplus 00011=11011\;,$$

where $\oplus$ represents bitwise addition mod $2$, and

$$\bigwedge_{k=1}^3a_k=01010\otimes 11011\otimes 00011=00010\;,$$

where $\otimes$ represents bitwise multiplication.