Let A={0,1,2,3,4,5}. Let R be a relation on A defined by (x,y)∈R if and only if max{x,y}∈{3,4}. Then among the statements
(S₁): The number of elements in R is 18, and
(S₂): The relation R is symmetric but neither reflexive nor transitive
Which of the following is correct?
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Step-by-Step Solution
Step 1: Determine the elements of the relation R
The relation R is defined on the set A={0,1,2,3,4,5}. A pair (x, y) belongs to R if the maximum of x and y is either 3 or 4. We need to list all such pairs.
Step 2: List pairs where max(x, y) = 3
If max{x,y}=3, then both x and y must be less than or equal to 3, and at least one of them must be 3. The possible pairs are (0,3),(1,3),(2,3),(3,3),(3,0),(3,1),(3,2). There are 7 such pairs.
Step 3: List pairs where max(x, y) = 4
If max{x,y}=4, then both x and y must be less than or equal to 4, and at least one of them must be 4. The possible pairs are (0,4),(1,4),(2,4),(3,4),(4,4),(4,0),(4,1),(4,2),(4,3). There are 9 such pairs.
Step 4: Calculate the total number of elements in R
The total number of elements in R is the sum of the pairs where max{x,y}=3 and the pairs where max{x,y}=4. So, ∣R∣=7+9=16. Therefore, statement (S \textsubscript1) which claims ∣R∣=18 is incorrect.
Step 5: Check properties of the relation R
1. Reflexivity: For R to be reflexive, (a,a) must be in R for all a∈A. However, for a=0, max{0,0}=0∈/{3,4}. So (0,0)∈/R, which means R is not reflexive.
2. Symmetry: If (x,y)∈R, then max{x,y}∈{3,4}. Since max{x,y}=max{y,x}, it follows that max{y,x}∈{3,4}, which means (y,x)∈R. Thus, R is symmetric.
3. Transitivity: For R to be transitive, if (x,y)∈R and (y,z)∈R, then (x,z)∈R. Consider (2,3)∈R (since max{2,3}=3) and (3,0)∈R (since max{3,0}=3). However, (2,0)∈/R because max{2,0}=2∈/{3,4}. Therefore, R is not transitive.
Statement (S \textsubscript2) is correct.