Answer to Question #198875 in Discrete Mathematics for Vina

Question

The relation P does not satisfy trichotomy. Which ordered pairs should be included in P so that an extended relation P1 (say) would satisfy trichotomy? (For trichotomy, each element of B must be paired with each other different element in B to form elements of P1. For example, we see that b  {a} but neither (b, {a}) nor ({a}, b) are elements of P, but at least one of these elements should be an element of P1. We include (b, {a}) in P1: P1 = {(a, b), (b, {a, b}), ({a, b}, a), ({b}, a), (a, {a}), (b, {a}), ...}.

Choose the alternative that provides all the missing ordered pairs that should be included as elements of P1 in order for P1 to be a relation that satisfies trichotomy. 1. (b, {a}), (b, {b}), (b, a), ({a}, {a, b}) & ({a, b}, {b})

2. ({a}, b), (b, {b}), ({b}, {a, b}) & ({a, b}, {a})

3. (b, {a}), ({b}, b), ({a}, {b}), ({b}, {a, b}) & ({a, b}, {a})

4. (b, {a}), (b, {b}), ({a}, {b}) & ({a, b}, {a, b}) 

Answer 1

set B = {a, b, {a}, {b}, {a, b}}

P = {(a, b), (b, {a, b}), ({a, b}, a), ({b}, a), (a, {a})}

for all x, y $\isin$ B with x ≠ y, we have (x, y) $\isin$ P₁ or (y, x)$\isin$ P₁

Answer: 2. ({a}, b), (b, {b}), ({b}, {a, b}) & ({a, b}, {a})