Get the complete, step-by-step math solution for: "Let {X}= {R} × {R}. Define a relation R on X as: a_1, b_1 R a_2, b_2 b_1=b_2. Statement I: {R} is an equivalence relation. Statement II: For some (a, ...". Powered by SolveForX AI math tutor.
Step 1: Analyze Statement I for Equivalence Relation Properties
To determine if R is an equivalence relation, we must check for reflexivity, symmetry, and transitivity. A relation R on a set X is reflexive if (a,a)∈R for every a∈X. It is symmetric if whenever (a,b)∈R, then (b,a)∈R. It is transitive if whenever (a,b)∈R and (b,c)∈R, then (a,c)∈R. In our case, the relation is defined as (a1,b1)R(a2,b2)⇔b1=b2. We can see that b=b is always true, so it's reflexive. If b1=b2, then b2=b1, so it's symmetric. If b1=b2 and b2=b3, then b1=b3, so it's transitive. Since all three properties hold, R is an equivalence relation.
Step 2: Analyze Statement II for the set S
Statement II describes a set S based on the relation R. Given (a,b)∈X, the set S consists of all points (x, y) such that (x, y) R (a, b). According to the definition of R, this means y=b. Therefore, the set S represents all points (x, y) where the y -coordinate is fixed at b, while x can be any real number. This describes a horizontal line.
Step 3: Compare the line S with y=x
The set S represents a horizontal line with equation y=b. A line parallel to y=x would have a slope of 1. A horizontal line has a slope of 0. Since the slopes are different (0=1), a horizontal line y=b is not parallel to the line y=x. Therefore, Statement II is false.
Step 4: Conclusion
Based on our analysis, Statement I, which claims R is an equivalence relation, is true. Statement II, which claims the set S represents a line parallel to y=x, is false because S represents a horizontal line y=b, which is not parallel to y=x.