Worksheet Sets - 7

​Sets: Comprehensive Web Worksheet

​Q. No. 1 [Focus: K, U, A]

​During a school club registration at a secondary school in Pokhara, the math club coordinator lists two separate student groups: Group A = \{\text{Ram}, \text{Sita}, \text{Hari}\} and Group B = \{\text{Sita}, \text{Hari}, \text{Ram}\}.

• ​a) Define what makes two sets equal sets based on their elements. [1K]
• ​b) Examine Group A and Group B. Are these two sets equal, equivalent, or both? Give a structural reason for your classification. [1U]
• ​c) If a third group, Group C, has a cardinal number of n(C) = 3 but contains completely different students, what is the exact mathematical relationship between Group A and Group C? [1A]

​Q. No. 2 [Focus: K, U, HA]

​Sita is organizing her home library bookshelf in Syangja. She creates Set M, which represents "the set of all talking monkeys living in Nepal," and Set P, which contains "the current Prime Minister of Nepal."

• ​a) Write down the specific mathematical name and the standard symbol used to denote a set like Set M that contains no elements. [1K]
• ​b) What is the special name given to a set containing exactly one element, like Set P? Write down its cardinal number n(P). [1U]
• ​c) Sita's brother claims that because a null set has no elements, it cannot be written inside curly brackets, and its cardinal number is the same as a singleton set. Critically evaluate his statement and correct any conceptual errors. [1HA]

​Q. No. 3 [Focus: K, A, HA]

​Anil sir asks his Class 8 students to analyze two mathematical collections: Set X = \{x : x \text{ is a prime factor of } 10\} and Set Y = \{y : y \text{ is a multiple of } 10\}.

• ​a) Define the core operational difference between a finite set and an infinite set. [1K]
• ​b) List Set X and Set Y in roster (tabular) form, and explicitly state which one is finite and which one is infinite. [1A]
• ​c) If we create a new Set Z containing all positive multiples of 10 that are less than 5, classify the type of set formed and deduce its cardinality n(Z). [1HA]

​Q. No. 4 [Focus: K, U, A]

​A local cooperative in Ghandruk tracks the livestock owned by small farmers. They define the Universal Set U as all domestic animals in the village. Farmer Pasang owns a subset of animals represented by P = \{\text{Yak}, \text{Cow}, \text{Goat}\}.

• ​a) Define what a Universal Set represents in any given mathematical context. [1K]
• ​b) If Pasang sells all her animals, her asset set becomes empty (\emptyset). Explain why an empty set is structurally considered a subset of Pasang's original set P. [1U]
• ​c) Write down all the possible individual subsets of Pasang’s livestock set P. [1A]

​Q. No. 5 [Focus: K, A, HA]

​Bikram is designing a digital inventory program for a traditional handicraft shop in Bhaktapur. He works with a specific set of clay pottery molds, Set S = \{\text{Pot}, \text{Vase}, \text{Plate}\}.

• ​a) Write down the fundamental mathematical formula used to calculate the total number of subsets for any given finite set with n elements. [1K]
• ​b) Using the formula, calculate the total number of subsets that can be generated from Bikram's inventory set S. [1A]
• ​c) If Bikram introduces a new item to the set, making the total number of subsets jump to 16, deduce how many elements are now in the updated set. [1HA]

​Q. No. 6 [Focus: K, U, A]

​Maya is categorizing traditional musical instruments for a cultural museum display in Lalitpur. She sets up a main collection M = \{\text{Madal}, \text{Flute}, \text{Sarangi}\} and a smaller display case D = \{\text{Madal}, \text{Sarangi}\}.

• ​a) State the mathematical condition required for Set D to be called a proper subset of Set M. [1K]
• ​b) Explain the conceptual difference between a proper subset and an improper subset using the collections M and D as a direct reference. [1U]
• ​c) Write down the specific formula for calculating the number of proper subsets of a set having n elements, and calculate the total number of proper subsets for Maya's main collection M. [1A]

​Q. No. 7 [Focus: U, A, HA]

​Kiran is compiling a list of vowels used in writing a specific local dialect framework. He creates a reference set V = \{a, e, i, o, u\}.

• ​a) Identify the total number of elements n(V) and explain why the set V itself is classified as an improper subset of V. [1U]
• ​b) Calculate the total number of proper subsets and the total number of improper subsets that can be formed from Set V. [1A]
• ​c) Kiran claims that if he removes two vowels from his set, the number of proper subsets will decrease by exactly 24. Prove mathematically whether his claim is true or false. [1HA]

​Q. No. 8 [Focus: K, U, A]

​Rajesh is setting up a sports tournament schedule for local clubs in Chitwan. He tracks Set T = \{x : x \text{ is a day of the week starting with the letter 'K'}\}.

• ​a) What is the cardinal number of a null set? [1K]
• ​b) Translate Rajesh's set T from set-builder notation into roster form and explain why it fits the definition of a null set. [1U]
• ​c) If Rajesh changes the rule to "days of the week starting with 'S'", write down the new set in roster form and calculate its cardinality. [1A]

​Q. No. 9 [Focus: K, A, HA]

​Dolma operates an organic apple orchard packaging unit in Mustang. She defines a quality testing batch as Set A = \{1, 2, 3, 4\}.

• ​a) State the rule for finding the number of improper subsets for any non-empty finite set. [1K]
• ​b) List all the proper subsets of Dolma's quality testing batch A that contain exactly 3 elements. [1A]
• ​c) If an unknown packaging Set B has exactly 127 proper subsets, work backward to determine the cardinal number n(B) of that set. [1HA]

​Q. No. 10 [Focus: U, A, HA]

​Niranjan is creating a classification system for geometric shapes on his educational website. He sets up a universal tracking framework where Set U = \{\text{all quadrilaterals}\}, Set R = \{\text{all rectangles}\}, and Set S = \{\text{all squares}\}.

• ​a) Describe the subset relationship between Set R and Set S. Which one is a subset of the other? Explain why. [1U]
• ​b) If a student claims that Set R is an improper subset of Set U, use the definition of improper subsets to calculate why this statement is false. [1A]
• ​c) Suppose we define a new set W = \{x : x \text{ is a triangle with } 4 \text{ sides}\}. Analyze the properties of set W, determine its cardinality, and explain how it relates to the Universal Set U. [1HA]