Let A={−2,−1,0,1,2,3}. Define a relation R on A by xRy if and only if y=max(x,1). If l is the number of elements in R, and m and n are the minimum numbers of ordered pairs that must be added to R to make it reflexive and symmetric respectively, then l+m+n is equal to:
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Step-by-Step Solution
Step 1: Determine the relation R and its size l
The relation R is defined for elements x, y in set A such that y is the maximum of x and 1. We need to find all ordered pairs (x,y) that satisfy this condition for each x∈A. We will then count the number of elements in R, which is l.
Step 2: Calculate the elements of R
For each x∈A:
- If x=−2, max(−2,1)=1, so (−2,1)∈R.
- If x=−1, max(−1,1)=1, so (−1,1)∈R.
- If x=0, max(0,1)=1, so (0,1)∈R.
- If x=1, max(1,1)=1, so (1,1)∈R.
- If x=2, max(2,1)=2, so (2,2)∈R.
- If x=3, max(3,1)=3, so (3,3)∈R.
Thus, R={(−2,1),(−1,1),(0,1),(1,1),(2,2),(3,3)}. The number of elements in R is l=6.
Step 3: Determine m for reflexivity
For a relation to be reflexive, every element x∈A must have (x,x)∈R. The set A={−2,−1,0,1,2,3}. The reflexive pairs are (−2,−2),(−1,−1),(0,0),(1,1),(2,2),(3,3).
From R, we have (1,1),(2,2),(3,3). The missing reflexive pairs are (−2,−2),(−1,−1),(0,0).
Therefore, m=3 pairs must be added to make R reflexive.
Step 4: Determine n for symmetry
For a relation to be symmetric, if (x,y)∈R, then (y,x) must also be in R. We check each pair in R:
- (−2,1)∈R, but (1,−2)∈/R. So, (1,−2) must be added.
- (−1,1)∈R, but (1,−1)∈/R. So, (1,−1) must be added.
- (0,1)∈R, but (1,0)∈/R. So, (1,0) must be added.
- (1,1)∈R, and (1,1)∈R. No addition needed.
- (2,2)∈R, and (2,2)∈R. No addition needed.
- (3,3)∈R, and (3,3)∈R. No addition needed.
Therefore, n=3 pairs must be added to make R symmetric.
Step 5: Calculate l + m + n
We have found l=6, m=3, and n=3. Now we sum these values to get the final answer.