Which of the following statements are true? Justify your answers. (This means that if you think a statement is false, give a short proof or an example that shows it is false. If it is true, give a short proof for saying so. For instance, to show that ‘{1, padma, blue} is a set’ is true, you need to say that this is true because it is a well-defined collection of 3 objects.)
i) The collection of Venn diagrams is a set.
iii) If z ∈ R , then | z | = z .
vi) Any system of three linear equations in two variables has no solution.
vii) If a matrix has n2 entries, where n ∈ N , then it is a square matrix.
viii) For n > 2 , the AM of the first n natural numbers is greater than n + 1.
Let's go through each statement and determine whether it is true or false, providing a justification for each one.
i) The collection of Venn diagrams is a set.
- True: A Venn diagram represents a well-defined way of illustrating the relationships between sets. Each Venn diagram can be considered as a distinct object, and together, they form a collection that can be treated as a set. Therefore, this statement is true.
ii) (A \ B) ∪ C = A \ (B ∩ C) for any three sets A, B, and C.
- True: This statement is true. You can prove it using set theory laws. Let's break it down step by step:
Left-hand side (LHS): (A \ B) ∪ C
- This means you first take the elements in A that are not in B (A \ B) and then union that with C.
Right-hand side (RHS): A \ (B ∩ C)
- Here, you first find the intersection of B and C (B ∩ C), and then take the elements in A that are
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Apply De Moivre’s theorem to write (sqrt(3 + 1)5 in the form a + ib , with a, b ∈ R
Which of the following statements are true? Justify your answers. (This means that if you think a statement is false, give a short proof or an example that shows it is false. If it is true, give a short proof for saying so. For instance, to show that ‘{1, padma, blue} is a set’ is true, you need to say that this is true because it is a well-defined collection of 3 objects.)
i) The collection of Venn diagrams is a set.
iii) If z ∈ R , then | z | = z .
vi) Any system of three linear equations in two variables has no solution.
vii) If a matrix has n2 entries, where n ∈ N , then it is a square matrix.
viii) For n > 2 , the AM of the first n natural numbers is greater than n + 1.
Obtain the polar and exponential representations of z1 , z2 and z1z 2 , where zi = 1/2 - 2i and z2 = 3 + i
Solve by substitution, the system you have obtained in 5(a) .
Find the sum of the 5th roots of unity.
Is Cramer’s Rule applicable for solving the linear system below? If yes, apply it.
Otherwise, alter the last equation in the system so that the solution can be obtained
by applying the Rule.
x + y + z = π
− πx + πy +sqrt(2z) = 0
π2 x + π2 y + 2z = 0 .