For a ∈ ℂ , let A = z ∈ ℂ : Re ( a + z ¯ ) > Im ( a ¯ + z ) and B = z ∈ ℂ : Re ( a + z ¯ ) < Im ( a ¯ + z ) . Then among the two statements: ( S 1 ) : If Re ( a ) , Im ( a ) > 0 , then the set A contains all the real numbers ( S 2 ) : If Re ( a ) , Im ( a ) < 0 , then the set B contains all the real numbers,

Step-by-step solution

Let a = x 1 + i y 1 ,   a ¯ = x 1 - i y 1 And z = x 2 + i y 2 ,   z ¯ = x 2 - i y 2 Now, a + z ¯ = x 1 + x 2 + i y 1 - y 2 and a ¯ + z = x 1 + x 2 + i y 2 - y 1 Now according to the question, Re ( a + z ¯ ) = x 1 + x 2 and Im ( a ¯ + z ) = y 2 - y 1 A = z : x 1 + x 2 > y 2 - y 1 = z : x 1 + y 1 + x 2 > y 2 B = z : x 1 + x 2 < y 2 - y 1 = z : x 1 + y 1 + x 2 < y 2 If y 2 = 0 and x 1 , y 1 > 0 then A = z : x 2 > - x 1 + y 1 A covers a part of negative re