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Use the the Cartesian product X × Y formula.
The Cartesian product X × Y of sets X , Y is the set of all ordered pairs ( x , y ).
Two ordered pairs ( x1 , y1 ), ( x2 , y2 ) .
The Cartesian product X × Y of sets X , Y is the set of all ordered pairs ( x , y ).
Two ordered pairs ( x1 , y1 ), ( x2 , y2 ) .
Given that P = { 1, 2, 3 } and Q = { 4 , 5 } then,
P x Q = { 1, 2, 3 } x { 4 , 5 }
⇒ P x Q = { 1 } x { 4 , 5 } , { 2 } x { 4 , 5 } , { 3 } x { 4 , 5 }
⇒ P x Q = { 1 , 4 } , { 1 , 5 } , { 2 , 4 } , { 2 , 5 } , { 3 , 4 }, { 3 , 5 }
⇒ P x Q = { (1 , 4 ) , ( 1 , 5 ) , ( 2 , 4 ) , ( 2 , 5 ) , ( 3 , 4 ), ( 3 , 5) }
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A ∩ B = { a, b, c } ∩ { c, d, e, f }
A ∩ C = { a, b, c } ∩ { c, d, e }
A ∩ B = {a, b, c} ∩ {c, d, e, f}
A ∩ B = { c }
A ∩ C = { a, b, c } ∩ { c, d, e }
A ∩ C = { c }
∴ (A ∩ B) ∪ (A ∩ C) = { c }.
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U ∪ A = {a, b, c, d, e, f} ∪ {a, b, c}
U ∪ A = {a, b, c, d, e, f} ∪ {a, b, c} = {a, b, c, d, e, f} = U
(U ∪ A)′ = Φ.
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