Consider a set U of 23 different compounds in a chemistry lab. There is a subset S of U of 9 compounds, each of which reacts with exactly 3 compounds of U.
Consider the following statements:
Each compound in U \ S reacts with an odd number of compounds.
At least one compound in U \ S reacts with an odd number of compounds.
Each compound in U \ S reacts with an even number of compounds.
Which one of the above statements is ALWAYS TRUE?
A. Only I
B. Only II
C. Only III
Answer given in book is- The sum of the degrees of all the vertices in a graph is equal to twice the number of edges.
Here, Set S has 9 elements and each element has a degree of 3 as each element in S reacts to exactly 3 elements. So 9*3 + Sum. of degrees of vertices in U-S = 2* e
Since 2*e is even no, Sum of degrees of vertices in U-S is odd .
Thus,some vertex in U-S must have odd degree. .At least one compound in U-S reacts with an odd number of compounds. Option B is the correct answer.
But How to solve it using set theory? Also, is there any other way solve this question!