Consider the relation R on the set {−2,−1,0,1,2} defined by (a,b)∈R if and only if 1+ab>0. Then, among the statements:
I. The number of elements in R is 17
II. R is an equivalence relation
Solutions · 1
Official solution
Let S={−2,−1,0,1,2}. Condition: ab>−1.
- a=−2: need −2b>−1⇒b<0.5⇒b∈{−2,−1,0}: 3
- a=−1: need −b>−1⇒b<1⇒b∈{−2,−1,0}: 3
- a=0: always true: 5
- a=1: need b>−1⇒b∈{0,1,2}: 3
- a=2: need 2b>−1⇒b∈{0,1,2}: 3
n(R)=3+3+5+3+3=17. Statement I is true.
Equivalence check: Reflexive holds since 1+a2≥1>0 always. Symmetric holds since ab=ba. Transitivity fails: take a=2,b=0,c=−2. Then ab=0>−1 and bc=0>−1 , but ac=−4, so 1+ac=−3>0 . So R is not transitive, hence not an equivalence relation. Statement II is false.