(a)
(i) Copy and complete the addition \(\oplus\) and multiplication \(\otimes\) tables in modulo 5 on the set {2, 3, 4}.
| \(\oplus\) | 2 | 3 | 4 |
| 2 | |||
| 3 | |||
| 4 |
| \(\otimes\) | 2 | 3 | 4 |
| 2 | |||
| 3 | |||
| 4 |
(ii) Use the tables to:
(a) solve the equation 4 \(\otimes\)e\(\oplus\) 2 \(\equiv\) 1 (mod 5):
(b) find the value of n, if 4 \(\oplus\)n\(\otimes\) 2 \(\equiv\) (mod 5).
(b) Consider the following statements:
p: Landi has cholera,
q: Landi is in the hospital.
If p = q, state whether or not the following statements are valid:
(i) If Landi is in the hospital, then he has cholera.
(ii) If Landi is not in the hospital, then he does not have cholera.
(iii) If Landi does not have cholera, then he is not in the hospital.
Explanation
(a). Completing the table as follows
| \(\oplus\) | 2 | 3 | 4 |
| 2 | 4 | 0 | 1 |
| 3 | 0 | 1 | 2 |
| 4 | 1 | 2 | 3 |
| \(\otimes\) | 2 | 3 | 4 |
| 2 | 4 | 1 | 3 |
| 3 | 1 | 4 | 2 |
| 4 | 3 | 2 | 1 |
(a) (ii) (a), they obtained the value of e as 1.
(a) (ii) (b), the value of n was 2.
(b), they couldn't answer this question correctly. If p = q, then:
(i) q \(\Rightarrow\) p, this is invalid
(ii) \(sin\) q \(\Rightarrow\) \(sin\) p, this is valid
(iii) \(sin\) p \(\Rightarrow\) \(sin\) q, this is invalid