MAT1830 - Discrete Mathematics for Computer Science Assignment #2
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MAT1830 - Discrete Mathematics for Computer Science
Assignment #2
(1) Prove by simple induction that, for each integer n ≥ 2,
(1 − 
) × (1 − 
) × (1 − 
) × · · · × (1 − 
) = 
. [6]
(2) Let S1,S2 ,S3 ,S4 , . . . be the sequence of sets defined by S1 = {0, 1, 3, 4}, S2 = {1, 2, 4, 5} and Si = (Si−1 n {i + 3})△(Si−2 − {i − 3}) for each integer i ≥ 3.
Prove by strong induction that Sn = {n − 1,n,n + 2,n + 3} for each integer n ≥ 1. [7]
(3) Let R, S and T be sets defined as follows.
R = Z − {3, 7, 9}
S = {7, 8, 9, 10}
T = {x ∈ Z : x ≥ 8}
Find the following.
(i) R ∩ S
(ii) T − S
(iii) (S ∩ T) × (S − R)
(iv) {X ∈ (P(S) − P(T)) : |X| = 2}
(v) 'P(T − R) × P(S)'
[No explanation required.] [5]
(4) Let X and Y be sets. Select which one of the following statements is equivalent to the statement
P ((X ∪ Y) × Y) ⊆ P (X × (X ∪ Y)).
(a) “X = Y”
(b) “X ⊆ Y”
(c) “Y ⊆ X”
(d) “X ⊆ Y or Y ⊆ X”
(e) “X = Y = ∅”
(f) “X = ∅”
(g) “Y = ∅”
(h) “X = ∅ or Y = ∅”
(i) “X ∩ Y = ∅”
[No explanation required.] [2]
2023-05-26