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For a ∈ C, let A = {z ∈ C : Re(a + z̄) > Im(ā + z)} and B = {z ∈ C : Re(a + z̄) < Im(ā + z)}. Then among the two statements : (S1) : If Re(a), Im(a) > 0, then the set A contains all the real numbers (S2) : If Re(a), Im(a) < 0, then the set B contains all the real numbers

1. Only (S2) is true
2. Only (S1) is true
3. Both are true
4. Both are false

Correct answer: Both are false

Solution

Both statements are false because the conditions given for the real parts and imaginary parts of 'a' do not lead to the specified sets A and B containing all real numbers. Specifically, the inequalities derived from the definitions of A and B do not hold for all real numbers under the given conditions.

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