Extra Content Added

Author: aryan

content/notes/iit-madras/data-science-and-application/foundational-level/mathematics/week 1/aq1.1.md

@@ -8,15 +8,19 @@ excludeSearch: false
 weight: 1.5
 ---
 
-1. Below is a list of numbers: 22, -17, 47, -2000, 0, 1, 43, 1729, 6174, -63, 100, 32, -9. How many natural numbers are there in the given list?
+{{< border >}}
 
-6
+**1**. Below is a list of numbers: 22, -17, 47, -2000, 0, 1, 43, 1729, 6174, -63, 100, 32, -9. How many natural numbers are there in the given list?
 
-7
+a) 6
 
-8
+b) 7
+
+c) 8
+
+d) 9
+{{< /border >}}
 
-9
 {{< border type="tip" title="Solution" >}}
 Based on the sources, the set of **natural numbers** is denoted by **N**. This set includes **0, 1, 2, 3, 4, 5, and so on**. The sources explicitly state that whenever they are talking about natural numbers, it **always includes a 0**, even though some books may not. Natural numbers are primarily used for counting.
 
@@ -44,15 +48,18 @@ Counting these numbers, we find there are **9** natural numbers in the given lis
 This corresponds to one of the options provided. The number of natural numbers in the list is **9**.
 {{< /border >}}
 
-2. Suppose a class has M students. If we distribute 3 pens to each student, then 4 pens will remain. If there are 70 pens in total, what is the value of M?
+{{< border >}}
+**2**. Suppose a class has M students. If we distribute 3 pens to each student, then 4 pens will remain. If there are 70 pens in total, what is the value of M?
 
-22
+a) 22
 
-20
+b) 20
 
-19
+c) 19
+
+d) 23
+{{< /border >}}
 
-23
 {{< border type="tip" title="Solution" >}}
 Okay, let's determine the value of M, the number of students in the class, based on the information provided and drawing on the concepts discussed in the sources.
 
@@ -89,15 +96,18 @@ Let's verify this: If there are 22 students and each gets 3 pens, that's 22 * 3
 The value of M is **22**. This is one of the options provided.
 {{< /border >}}
 
-3. Consider ‘2’ to be the first prime number. Which of the following will be the sixteenth prime number?
+{{< border >}}
+**3**. Consider ‘2’ to be the first prime number. Which of the following will be the sixteenth prime number?
 
-47
+a) 47
 
-61
+b) 61
 
-59
+c) 59
+
+d) 53
+{{< /border >}}
 
-53
 {{< border type="tip" title="Solution" >}}
 Based on the sources and our conversation, we can determine the sixteenth prime number.
 
@@ -173,21 +183,24 @@ Comparing this to the provided options:
 Therefore, the sixteenth prime number is **53**.
 {{< /border >}}
 
-4. If 
+{{< border >}}
+**4**. If 
 x
 x mod 3 = 1 and 
 x
 x mod 7 = 4. Which of the following is(are) possible values for 
 x
 x?
 
-25
+a) 25
+
+b) 34
 
-34
+c) 58
 
-58
+d) 67
+{{< /border >}}
 
-67
 {{< border type="tip" title="Solution" >}}
 Based on the sources and our conversation history, we can evaluate which of the given options are possible values for `x` by using the concept of the **modulo** operation.
 
@@ -226,8 +239,8 @@ Let's check each of the possible values provided:
 Therefore, the possible values for `x` among the given options are **25** and **67**.
 {{< /border >}}
 
-5. 
-Suppose Sheetal’s birthday is on the 
+{{< border >}}
+**5**. Suppose Sheetal’s birthday is on the 
 P
 Pth of October and Karthik’s birthday is on the 
 Q
@@ -241,6 +254,8 @@ P
 2
 Q
 P−2Q will be __________
+{{< /border >}}
+
 {{< border type="tip" title="Solution" >}}
 Based on the information from the sources, let's determine the value of P - 2Q.
 

content/notes/iit-madras/data-science-and-application/foundational-level/mathematics/week 1/aq1.10.md

@@ -8,23 +8,151 @@ excludeSearch: false
 weight: 10.5
 ---
 
+**Answer the questions 1-3, based on following information:**
+
+Let A = {x|x ∈ ℕ, x < 10 and x is odd}
+
+B = {y|y ∈ ℕ, y is a perfect square and 15 < y < 40}
+
+{{< border >}}
+**Q1**. Which of the following is a subset of B × A?
+
+○ {(36, 3), (25, 5), (36, 6)}
+
+○ {(1, 25), (6, 36), (7, 25), (3, 36)}
+
+○ {(16, 5), (25, 9), (36, 3), (16, 1)}
+
+○ {(1, 16), (5, 16), (3, 36), (9, 36), (7, 25)}
+{{< /border >}}
+
+{{< border type="tip" title="Solution" >}}
+
+{{< /border >}}
+
+
+{{< border >}}
+**Q2**. Which of the following is a subset of A × A?
+
+○ {(1, 3), (3, 5), (5, 9), (7, 3), (7, 7)}
+
+○ {(36, 25), (25, 25), (36, 36), (16, 25)}
+
+○ {(3, 3), (1, 5), (3, 6), (9, 9)}
+
+○ {(3, 5), (5, 1), (9, 4), (1, 1), (7, 1)}
+{{< /border >}}
+
+{{< border type="tip" title="Solution" >}}
+
+{{< /border >}}
+
+
+{{< border >}}
+**Q3**. Which of the following is(are) element(s) of A × B × A?
+
+□ (2, 25, 3)
+
+□ (5, 25, 9)
+
+□ (3, 36, 9)
+
+□ (5, 16, 1)
+{{< /border >}}
+
+{{< border type="tip" title="Solution" >}}
+
+{{< /border >}}
+
+**Answer the questions 4-5, based on following information:**
+
+Suppose the Cartesian product M × M has 9 elements among which 2 elements are (2, 4) and (3, 2).
+
+{{< border >}}
+**Q4**. Which of the following sets is M?
+
+○ {9, 2, 4}
+
+○ {3, 2, 9}
+
+○ {2, 3, 4}
+
+○ {3, 4, 9}
+{{< /border >}}
 
 {{< border type="tip" title="Solution" >}}
 
 {{< /border >}}
 
+
+{{< border >}}
+**Q5**. Remaining elements of M × M are
+
+○ {(2, 3), (2, 2), (3, 4), (3, 2), (4, 4), (4, 2), (3, 3)}
+
+○ {(2, 3), (2, 2), (3, 4), (4, 3), (4, 4), (4, 2), (3, 3)}
+
+○ {(2, 3), (2, 2), (4, 3), (2, 4), (4, 4), (4, 2), (3, 3)}
+
+○ {(2, 2), (2, 3), (4, 3), (3, 2), (3, 3), (4, 4), (3, 4)}
+{{< /border >}}
+
 {{< border type="tip" title="Solution" >}}
 
 {{< /border >}}
 
+**Answer the questions 6-8, based on following information:**
+
+Suppose the relation R = {(Snail, Frog), (Bird, Bird), (Fox, Frog), (Snail, Fox)} is defined on the set S = {Snail, Fox, Bird, Frog}.
+
+{{< border >}}
+**Q6**. The relation R on the set S is(are)
+
+○ Reflexive
+
+○ Symmetric
+
+○ Transitive
+
+○ None of these
+{{< /border >}}
+
 {{< border type="tip" title="Solution" >}}
 
 {{< /border >}}
 
+
+{{< border >}}
+**Q7**. Let P be a subset of (S × S)\R. What is the set of possible cardinalities of P?
+
+○ {1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12}
+
+○ {}
+
+○ {1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11}
+
+○ {0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12}
+{{< /border >}}
+
 {{< border type="tip" title="Solution" >}}
 
 {{< /border >}}
 
+
+{{< border >}}
+**Q8**. What is the minimum number of elements of the Cartesian product S × S that need to be added to R such that the relation becomes symmetric?
+
+○ 4
+
+○ 8
+
+○ 2
+
+○ 3
+{{< /border >}}
+
 {{< border type="tip" title="Solution" >}}
 
-{{< /border >}}
+{{< /border >}}
+
+