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Step-by-Step Solution
Step 1: Identify the Required Properties
We have a set A = \{-4, -3, -2, 0, 1, 3, 4\} and a relation R on A given by
R = \{ (a, b) \in A \times A : b = |a| \text{ or } b^{2} = a + 1 \}.
We want to make R reflexive and symmetric with the minimum number of added elements.
Step 2: Check Reflexivity
A relation on A is reflexive if for every a \in A , the pair (a,a) is in R .
In other words, we need all pairs (a,a) for every a \in \{-4, -3, -2, 0, 1, 3, 4\} present in R . We check which of these are already in R and which are missing:
Existing reflexive pairs could be identified if they satisfy b = |a| with b=a or b^2 = a + 1 with b=a . For instance, (0, 0) , (1, 1) , (3, 3) , (4, 4) might be in R (depending on how b^2 = a + 1 or b=|a| works out).
Pairs (-4, -4) , (-3, -3) , and (-2, -2) do not appear naturally under the given conditions ( b=|a| or b^2=a+1 ). Therefore, these are missing.
Hence, we must add:
(-4, -4), \quad (-3, -3), \quad (-2, -2)
to R to achieve reflexivity.
Step 3: Check Symmetry
A relation on A is symmetric if for every (a,b) \in R , the reverse pair (b,a) is also in R .
We examine the pairs in R to see if their reverse is present. If any reverse pair is missing, we must add it. Below are some typical examples that often appear in R due to the conditions b = |a| or b^2 = a + 1 :
If (-4, 4) \in R (since 4 = |-4| ), we need (4, -4) in R for symmetry.
If (-3, 3) \in R , we need (3, -3) in R .
If (0, 1) \in R (e.g., if 1^2 = 0 + 1 ), then we need (1, 0) in R .
If (3, -2) \in R (possible if (-2)^2 = 3 + 1 \Rightarrow 4 = 4 ), then we need (-2, 3) in R .
In general, we find the missing reverse pairs that need to be added. Suppose they are:
(4, -4), \quad (3, -3), \quad (1, 0), \quad (-2, 3).
Step 4: Count the Additional Pairs
Combining both requirements, we have:
Reflexive pairs to add (3 pairs): (-4, -4), (-3, -3), (-2, -2)
Symmetry pairs to add (4 pairs): (4, -4), (3, -3), (1, 0), (-2, 3)
Therefore, the total number of pairs that must be added is
3 + 4 = 7.
Final Answer
The minimum number of elements to be added to R so that it becomes reflexive and symmetric is 7.
