Exam Questions & Step-by-Step Markschemes (4 Problems)
Question 1 • 2023
5 Marks
Let \mathbb{N} denote the set of all natural numbers. Define a relation R on \mathbb{N} \times \mathbb{N} by: (a, b) R (c, d) \iff a + d = b + c Show that R is an equivalence relation. Also, find the equivalence class [(2, 5)] .
Answer: R \text{ is an equivalence relation}; \quad [(2, 5)] = \{(1, 4), (2, 5), (3, 6), (4, 7), \dots\}
Question 2 • 2023
4 Marks
Show that the relation R in the set A = \{x \in \mathbb{Z} : 0 \le x \le 12\} , given by: R = \{(a, b) : |a - b| \text{ is a multiple of } 4\} is an equivalence relation. Also, find the set of all elements related to 1 (i.e. the equivalence class [1] ).
Answer: R \text{ is an equivalence relation}, \quad [1] = \{1, 5, 9\}
Question 3 • 2024
3 Marks
Let L be the set of all straight lines in the xy -plane and R be the relation in L defined by R = \{(L_1, L_2) : L_1 \text{ is parallel to } L_2\} . Show that R is an equivalence relation. Find the set of all lines related to the line y = 2x + 4 .
Answer: y = 2x + c, \quad c \in \mathbb{R}
Question 4 • 2024
5 Marks
Show that the relation R in the set A = \{x \in \mathbb{Z} : 0 \le x \le 12\} , given by R = \{(a, b) : |a - b| \text{ is a multiple of } 4\} , is an equivalence relation. Find the set of all elements related to 1.
Answer: \text{Equivalence class of 1: } \{1, 5, 9\}