Initial work

CS1029/practical-2023-10-20/functions

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+1. a) f(x) is undefined for x = 0
+   b) f(x) is complex for x < 0
+   c) multivalued
+
+2. a) multivalued
+   b) It is
+   c) undefined at x = 2
+
+3. a) Z >= 0, {0..9}
+   b) 0.. -> 1..?
+   c) All bit strings -> N
+   d) All bit strings -> N
+   e) (Z^+)^2 -> Z^+
+   f) (Z^+)^2 -> Z^+
+
+4. a) 1 b) 2 c) -1 d) 0 e) 2 f) 3 g) 0 h) 2
+5. a) 1 b) 0 c) 0 d) -1 e) 3 f) -1 g) 2 h) 1
+
+8. a) 6 b) 24 c) 120 d) 3_628_800
+
+10. a)y b)n c)n
+11. {a}
+
+12. a) y b) n c) n d) n
+13. a) y b) n) c) y) d) n
+
+14. ynyy
+
+15. a) everyone has a phone number (phone numbers should already be uniquie)
+    b) None (student ids should already be unique)
+	c) Every student gets a unique grade
+	d) no two people come from the same town
+
+16. a) the set of phone numbers
+    b) the set of student ids
+	c) 1..100
+	d) all towns
+
+17. ynnn
+18. a) { 1 }
+    b) { -1, 1, 5, 9, 15 }
+	c) { 0, 1, 2 }
+	d) { 0, 1, 2 }
+
+19. f.g = x^2 + 4x + 5, g.f = x^2 + 3
+20. f + g = x^2 + x + 3
+    fg = x^3 + 2x^2 + x + 2
+
+21. f^-1 doesn't exist, assuming f^-1(x) = +sqrt(x)
+   a) { 1 }
+   b) { x | 0 < x < 1 }
+   c) { x | x > 2 }

CS1029/practical-2023-10-20/m.py

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+def floor(n):
+	return n - (n % 1)
+
+def ceil(n):
+	return n - (n % -1)
+
+
+assert floor(-1) == -1
+assert floor(1) == 1
+assert floor(1.2) == 1
+assert floor(-1.2) == -2
+assert floor(0) == 0
+assert floor(0.1) == 0
+assert floor(-0.1) == -1
+assert ceil(-1) == -1
+assert ceil(1) == 1
+assert ceil(1.2) == 2
+assert ceil(-1.2) == -1
+assert ceil(0) == 0
+assert ceil(0.1) == 1
+assert ceil(-0.1) == 0
+

CS1029/practical-2023-10-20/sequences

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+1. a) 3
+   b) -1
+   c) 787
+   d) 2639
+
+2. a) 128
+   b) 7
+   c) 2
+   d) -256
+
+3. a) 1, -2, 4, -8
+   b) 3, 3, 3, 3
+   c) 8, 13, 23, 71
+   d) 2, 0, 8, 0
+
+4. a) 2, 5, 8, 11, 14, 17, 20, 23, 26, 29
+   b) 0, 0, 0, 1, 1, 1, 2, 2, 2, 3
+   c) 1, 1, 3, 3, 5, 5, 7, 7, 9, 9
+   d) 3, 6, 12, 24, 48, 96, 192, 384, 768, 1336
+
+5. a_0 = 3, a_n = a_{n - 1} + 2 (i.e. the odd integers starting at 3)
+   the primes starting at three
+   - one more
+
+6. a) 2, 12, 72, 432, 2592
+   b) 2, 4, 16, 256, 65536
+   c) 1, 2, 5, 11, 26
+   d) 1, 1, 6, 27, 204
+
+7. a) a_n = -3a_{n-1} + 4a_{n-2} = -3(0) + 4(0) = 0 = a_n
+   b) a_n = -3(1) + 4(1) = 4 - 3 = 1 = a_n
+   c) a_n = -3((-4)^{n-1}) + 4((-4)^{n-2}) = -3(-4)(-4^{n-2}) + 4((-4)^{n-2}) = (-4)^{n-2}((-3)(-4) + 4) = (-4)^{n-1}(-3 - 1) = (-4)^n = a_n
+   d) a_n = -3(2*(-4)^{n-1} + 3) + 4(2 * (-4)^{n-2} + 3) = -6(-4)^{n-1} - 9 - 2(-4)^{n-1} + 12 = -8(-4)^{n-1} + 3 = 2(-4)^n + 3 = a_n
+
+8.