Answer to Question #155079 in Discrete Mathematics for Tanveer shahzad

Question

A binary operation € on three variables A, B and C define as return true if at least two true return us true then prove that OR operation on three variable is equivalent to which of the following ?

A€(B€C)        2) A€(B€¬C)   3) A€(¬B€C)     4) ¬A€(B€C)

5) A€(¬B€¬C)     6) ¬A€(B€¬C)    7) ¬A€(¬B€C)    8) ¬A€(¬B€¬C)

Answer 1

As the operations OR and € are symmetric, the formula expressing the first operation through the second must be symmetric. Only the 1st and 8th formulas are symmetric. Let's consider them.

1) €(true, false, false) = false, but OR(true, false, false) = true. Therefore, the first formula doesn't work.

8) €(¬true, ¬true, ¬true) = false, but OR(true, true, true)= true. Therefore, the eighth formula doesn't work too.

Answer. There is no formula among (1)-(8) that expressing the OR-operation through the operation €.