Sets : Relation And Function Assignment
3.1 Question 1
Let us consider the following question with different solutions. Choose the best solution and explain why you choose it.
Problem: Prove that the product of two odd numbers is also an odd number.
- An integer is an odd number if it is two times another integer plus 1. Therefore the product of two odd numbers is four times the product of two other integers plus two times the sum of the two other integers plus one. This shows that the product of two odd numbers is odd
- m = 2k + 1, n = 2l + 1
mn = 4kl + 2(k + l) + 1 = 2(2kl + k + l) + 1.
- Suppose n and m are two odd numbers, then m = 2k + 1 and n = 2l + 1 for some
k, l ? Z. We have
m n = (2k + 1)(2l + 1)
= 4kl + 2k + 2l + 1
= 2(2kl + k + l) + 1
= 2q + 1,
where q = 2kl + k + l.
We can see that q = 2kl + k + l Z as k, l Z. Therefore, m n = 2q + 1 is an odd number. In other words, the product of two odd numbers is also odd.
- Suppose n and m are two odd numbers, then m=2k+1 and n=2l+1 for some integers k and l. We have
mn=(2k+1)(2l+1)=4kl+2k+2l+1=2(2kl+k+l)+1=2q+1, where q=2kl+k+l.
We can see that q is an integer as k, l Z. Therefore, mn =2q+1 is an odd number. In other words, the product of two odd numbers is also odd.
3.2 Question 2
Consider the following problem and its solution.
- Problem:Calculate A100 where
A = 0 1 .
?1 ?1
- Solution:By a straightforward calculation, we obtain
A100 = 0 1 .
?1 ?1
This solution does not explain how we calculate A100. Therefore, this solution will be marked as ‘insufficient explanation’.
Can you provide an explanation to this solution to make it complete?
3.3 Question 3
Before using a theorem or lemma, make sure you (carefully) check and state if all conditions of the theorem or lemma are actually satisfied.
Let f (x) = tan x. In the following statements, choose the correct statement. Briefly explain your answer.
- Sincef (0) = f (?) = 0, by Rolle’s theorem there exists c ? (0, ?) such that f?(c) =
- Calculatingthe derivative of f , we have
f?(x) = 1 cos2(x)
= 1 + tan2 x.
We note that over the interval [0, ?], f is not defined at x = ?/2. As 1 + tan2 x > 0 for all
x ? [0, ?] and x /= ?/2, there does not exist c ? (0, ?) such that f?(c) = 0.
- Both(i) and (ii) are
- Both(i) and (ii) are
3.4 Question 4
Consider the following statement and explanation. Can you conclude that the statement is true?
- Statement:If p is prime then 2p ? 1 is also
- Explanation:The statement holds for p = 2, 3, 5, Therefore, the statement is true, i.e., if p is prime then 2p ? 1 is also prime.
3.5 Question 5
Consider the following problem and solution. Is the solution given below correct? If not, can you give a correct solution for the problem?
- Problem:Show that if A is a square matrix, then A + AT is
- Solution:For the case where A is a 1 × 1 matrix, let A = [a]. We have
A + AT = [a] + [a] = [2a],
which is symmetric.
For the case where A is a 2 × 2 matrix, let
A = a b .
c d
We have
A + AT = a b + a c = 2a b + c ,
which is symmetric.
c d b d
b + c 2d
Therefore, if A is a square matrix, then A + AT symmetric.
3.6 Question 6
Consider the following problem and choose all the valid solutions from the options given below.
- Problem:Prove that if ab = ac and a /= 0 then b = c for a, b, c ? R.