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)
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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 €.