CS103X: Discrete Structures Homework Assignment 1: Solutions

Exercise 1 (10 Points). Prove or give a counterexample for each of the following: (a) If A ⊆ B and B ⊆ C, then A ⊆ C. (b) If A ∈ B and B ∈ C, then A ∈ C. Solution: (a) Consider any element a ∈ A. Since A ⊆ B every element of A is also an element of B, so a ∈ B. By same reasoning, a ∈ C since B ⊆ C. Thus every element of A is an element of C, so A ⊆ C. (b) Let A = {1}, B = {1, {1}}, and C = {{1, {1}}, 3, {4}}. These sets satisfy A ∈ B and B ∈ C, but A / ∈ C. Exercise 2 (10 Points). If a(t), b(t), and c(t) are the lengths of the three sides of a triangle t in non-decreasing order (i.e. a(t) ≤ b(t) ≤ c(t)), we define the sets: • X := {Triangle t : a(t) = b(t)} • Y := {Triangle t : b(t) = c(t)} • T := the set of all triangles Using only set operations on these three sets, define: (a) The set of all equilateral triangles (all sides equal) (b) The set of all isosceles triangles (at least two sides equal) (c) The set of all scalene triangles (no two sides equal) Solution: (a) We require a(t) = b(t) and b(t) = c(t) (this obiviously implies a(t) = c(t)), so the set is X ∩ Y (b) An isoceles triangle t can have 1. a(t) = b(t), or 2. b(t) = c(t), or 3. a(t) = c(t).