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) for any three sets B,A and C .
iii)
iv) , has a root in R only if m is an odd number.
v)
vi) Any system of three linear equations in two variables has no solution
vii) If a matrix has 2 n entries, where n ∈ N , then it is a square matrix.
viii) For n > 4 , the AM of the first n natural numbers is greater than n +1
Let’s go through each of the eight statements manually, verifying whether they're true or false, with short reasoning or counterexamples: --- ### i) The collection of Venn diagrams is a set. Answer: TRUE Reason: A *set* is a well-defined collection of objects. The term "Venn diagrams" refers to a class of visual representations of sets and their relationships. Thus, "the collection of all Venn diagrams" is well-defined, even if infinite. --- ### ii) $(A \setminus B) \cup C = A \setminus (B \cap C)$ for any three sets A, B, and C. Answer: FALSE Counterexample: Let * $A = \{1, 2, 3\}$, * $B = \{2, 3\}$, * $C = \{3, 4\}$ Then: * $A \setminus B = \{1\}$ * $(A \setminus B) \cup C = \{1\} \cup \{3, 4\} = \{1, 3, 4\}$ Now compute RHS: * $B \cap C = \{3\}$ * $A \setminus (B \cap C) = A \setminus \{3\} = \{1, 2\}$ Since $\{1, 3, 4\} \ne \{1, 2\}$, the two sides are not equal. --- ### iii) If $z \in \mathbb{R}$, then $|z| = z$. Answer: FALSE Counterexample: Let $z = -5$. Then $|z| = 5 \ne -5 = z$. So $|z| = z$ ___ ___ ___ __________ ____ _________ ________ ___ _________ ____ ________.
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समीकरण का विविक्तकर प्राप्त कीजिए। इस तरह, इसके मूलों की प्रकृति बताइए।
egers Write an odd natural number as a sum of two int m and 1 m in a way that 2 m is maximum.
Prove that
मान लीजिए कि इस प्रकार है कि
तथा
है। तब, दर्शाइए कि
The annual bonus given to the employees of a company is 10% of their taxable incomes, after the state and central taxes are deducted. The state tax is 15% of taxable income. The central tax is 15% of taxable income after deducting the state tax. Formulate this situation for determining the bonus, as a linear system.
दिया रहने पर x और y ज्ञात कीजिए।
Give examples, with justification, of the following:
i) two non-zero, 3×3 matrices A and B , with A| = 0 |B| = 5/7 i;
ii) two non-singular 2× 2 matrices C and D , with |C| = √2| D |.