Question

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.

ii) (A \ B) ∪ C = A \ (B ∩ C) for any three sets A, B and C .

iii) If z ∈ R , then | z | = z .       

iv) xm + a1 xm-1 +.....+am-1x + am = 0,ai ∈ R ∀ i = 1,.... m , has a root in R only if m is an odd number.
v) x > 0 is necessary for x + 2 > 1 .

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.

02 Feb 2018
Answer :
Word Count : 733

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 not in that intersection.
   - The LHS and RHS are equivalent because

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