Question

Find

(a)\({\bf{N((}}\;{\bf{))}}\).

(b)\({\bf{N(}}\;{\bf{)))(}}\;{\bf{))((}}\;{\bf{)}}\).

(c)\({\bf{N(((}}\;{\bf{)((}}\;{\bf{))}}\).

(d)\({\bf{N(}}\;{\bf{)(((}}\;{\bf{)))((}}\;{\bf{)))}}\).

Short answer

(a) The value of\(N((\;))\)is 0.

(b) The value of\(N(\;)))(\;))((\;)\)is\( - 2\).

(c) The value of\(N(((\;)((\;))\)is\(2\).

(d) The value of \(N(\;)(((\;)))((\;)))\)is\( - 1\).

Step-by-step solution

Define Sets

The collection of the number of elements is define as set. The set is denoted by a variable that has number of elements inside the brackets. Also, the change in order or the repletion of the elements makes no change in the complete set.

Consider the following function on the set of strings of parentheses defined by:

\(N\left( \lambda \right) = 0,\;N((\;) = 1,\;N()) = - 1\)

(a) Find \({\bf{N((}}\;{\bf{))}}\)

The objective is to find the value of\(N((\;))\).

There is one left parentheses and one right parentheses more to\(N(\;)\).

Rewrite the function\(N((\;))\)as follows:

\(\begin{aligned}{l}N((\;)) &= N(() + N())\\\;\;\;\;\;\;\;\; &= 1 + ( - 1)\\\;\;\;\;\;\;\;\; &= 0\end{aligned}\)

Therefore, the value of \(N((\;))\) is 0.

(b) Find \({\bf{N(}}\;{\bf{)))(}}\;{\bf{))((}}\;{\bf{)}}\)

The objective is to find the value of\(N(\;)))(\;))((\;)\).

There are three left parentheses and five right parentheses more to\(N(\;)\).

Rewrite the function\(N(\;)))(\;))((\;)\)as follows:

\(\begin{aligned}{l}N(\;)))(\;))((\;) &= N())) + N()) + N(()\\\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\; &= ( - 2) + ( - 1) + 1\\\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\; &= - 2\end{aligned}\)

Therefore, the value of \(N(\;)))(\;))((\;)\) is\( - 2\).

(c) Find \({\bf{N(((}}\;{\bf{)((}}\;{\bf{))}}\)

The objective is to find the value of\(N(((\;)((\;))\).

There are four left parentheses and two right parentheses more to\(N(\;)\).

Rewrite the function\(N(((\;)((\;))\)as follows:

\(\begin{aligned}{l}N(((\;)(()) &= N(((\;) + N(())\\\;\;\;\;\;\;\;\;\;\;\;\;\; = N(((\;) + N()) + N(()\\\;\;\;\;\;\;\;\;\;\;\;\;\; &= 2 + ( - 1) + 1\\\;\;\;\;\;\;\;\;\;\;\;\;\; &= 2\end{aligned}\)

Therefore, the value of \(N(((\;)((\;))\) is\(2\).

(d) Find \({\bf{N(}}\;{\bf{)(((}}\;{\bf{)))((}}\;{\bf{)))}}\)

The objective is to find the value of\(N(\;)(((\;)))((\;)))\).

There are five left parentheses and six right parentheses more to\(N(\;)\).

Rewrite the function\(N(\;)(((\;)))((\;)))\)as follows:

\(\begin{aligned}{l}N(\;)(((\;)))((\;))) &= N(\;) + N((())) + N(()))\\\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\; = N(\;) + N())) + N((() + N(() + N()))\\\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\; = 0 + ( - 2) + 2 + 1 + ( - 2)\\\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\; &= - 1\end{aligned}\)

Therefore, the value of \(N(\;)(((\;)))((\;)))\)is\( - 1\).