Lisp programming – Physicspupil

Given two operands $a$ and $b$ which are real numbers, define the operators,

$a+b$ , $a-b$ , $a\times b$ , $a\div b$ , $a^{b}$ , $\sqrt[b]{a}$ , $a\bmod b$ ,

with respect to arithmetic operations such as addition, subtraction, multiplication, division, exponentiation, rooting, and Modulo.

+------+------------+-----+-----+-------------------------+
|  ID  |  operator  |  a  |  b  |  instance for example   |
+------+------------+-----+-----+-------------------------+
|  1   |  +         |  2  |  3  |  a + b  = 2 + 3    = 5  |
|  2   |  -         |  5  |  2  |  a - b  = 5 - 2    = 3  |
|  3   |  ×         |  3  |  2  |  a * b  = 3 * 2    = 6  |
|  4   |  ÷         |  9  |  3  |  a / b  = 9 / 3    = 3  |
|  5   |  ^         |  3  |  2  |  a ^ b  = 3 ^ 2    = 9  |
|  6   |  √         |  3  |  8  |  a √ b  = 3 √ 8    = 2  |
|  7   |  mod       |  7  |  4  |  a mod b = 7 mod 4 = 3  |
+------+------ -----+-----+-----+-------------------------+

+------+------------+-----+-----+-------------------------+

| ID | operator | a | b | instance for example |

+------+------------+-----+-----+-------------------------+

| 1 | + | 2 | 3 | a + b = 2 + 3 = 5 |

| 2 | - | 5 | 2 | a - b = 5 - 2 = 3 |

| 3 | × | 3 | 2 | a * b = 3 * 2 = 6 |

| 4 | ÷ | 9 | 3 | a / b = 9 / 3 = 3 |

| 5 | ^ | 3 | 2 | a ^ b = 3 ^ 2 = 9 |

| 6 | √ | 3 | 8 | a √ b = 3 √ 8 = 2 |

| 7 | mod | 7 | 4 | a mod b = 7 mod 4 = 3 |

+------+------ -----+-----+-----+-------------------------+

Not so accustomed to these, Lisp cannot interpret or compile them until we have defined those seven primitive functions, better known as symbolic computations; as were newborn infants taught human intelligence by grownup adults between homo sapiens.

To Lisp, arithmetic operations are usually $\textrm{\scriptsize{NOT}}$ predefined. We have in our toolkit $\textrm{\scriptsize{ONLY}}$ three mathematical functors, i.e., list, iteration, and recursion.

The roots of quadratic equation are given by the formula:

$\displaystyle{x=\frac{-b\pm\sqrt{b^2-4ac}}{2a}}$

the expected outcome of which is calculated by parts, first the discriminant ‘delta’:

(define (delta a b c) (- (square b) (* 4 a c)))

1	(define (delta a b c) (- (square b) (* 4 a c)))

$\Delta = b^2-4ac$ ;

then the plus-minus ‘pm’:

(define (pm u v)
	(values (cons (+ u v) (- u v))))

1 2	(define (pm u v) (values (cons (+ u v) (- u v))))

$\pm (x)=(+x,-x)$ ;

and hence

(/ (pm (- b) (sqrt (delta a b c))) (* a 2))

1	(/ (pm (- b) (sqrt (delta a b c))) (* a 2))

will solve for real roots, ignoring the complex though.

$\therefore$ Predefining the seven primitive functions is in order for solving any computational problems by Lisp.

(to be continued)

Implement the union-set operation for the unordered-list representation of sets.

_{Extracted from Structure and Interpretation of Computer Programs, SICP}

Attempts.

Define a function element-of-set?by

(define (element-of-set? x set)
(cond ((null? set) false)
((equal? x (car set)) true)
(else (element-of-set? x (cdr set)))))

(define (element-of-set? x set)

(cond ((null? set) false)

((equal? x (car set)) true)

(else (element-of-set? x (cdr set)))))

to determine whether an object xis an element of a set set.

Define a function intersection-set by

(define (intersection-set set1 set2)
(cond ((or (null? set1) (null? set2)) ’())
((element-of-set? (car set1) set2)
(cons (car set1)
(intersection-set (cdr set1) set2)))
(else (intersection-set (cdr set1) set2))))

(define (intersection-set set1 set2)

(cond ((or (null? set1) (null? set2)) ’())

((element-of-set? (car set1) set2)

(cons (car set1)

(intersection-set (cdr set1) set2)))

(else (intersection-set (cdr set1) set2))))

to draw the intersection intersection-set of two sets set1 and set2.

Define a function union-set by

(define (union-set set1 set2)
(cond ((and (null? set1) (null? set2)) ’())
((null? set1) set2)
((null? set2) set1)
((element-of-set? (car set1) set2)
(cons (car set1)
(union-set (cdr set1) set2)))
(else (union-set (cdr set1) set2))))

(define (union-set set1 set2)

(cond ((and (null? set1) (null? set2)) ’())

((null? set1) set2)

((null? set2) set1)

((element-of-set? (car set1) set2)

(cons (car set1)

(union-set (cdr set1) set2)))

(else (union-set (cdr set1) set2))))

to draw the union union-set of two sets set1 and set2.

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Category: Lisp programming

202205261415 Arithmetic in Lisp

202111251402 Exercise 2.59 in Lisp