Is This Correct Or Not? (Challenge)

Logic Level 2

We define the binary operator, \circ such that a a = a + 2 a \circ a = a+2 , a b = { a + 1 , if a > b b + 1 , if b > a a \circ b = \begin{cases} a+1 \quad , \quad \text{ if } a>b \\ b+1 \quad , \quad \text{ if } b>a \end{cases} .

Is this statement correct?

( x y = x + 1 ) OR ( x y = y + 1 ) ( x y x + 2 ) (x\circ y = x+1) \text{ OR } (x\circ y = y+1) \ne (x\circ y \ne x+2)

True False

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2 solutions

Kay Xspre
Feb 2, 2016

a b = { b + 1 , if a < b a + 2 , if a = b a + 1 , if a > b a \circ b = \begin{cases} b+1 \quad , \quad \text{ if } a<b \\ a+2 \quad , \quad \text{ if } a=b\\a+1 \quad , \quad \text{ if } a>b \end{cases}

If on the left-handed side, the result is either x + 1 x+1 or y + 1 y+1 , then x y x\neq y . If x y x \neq y , then the result on the right-handed side shall also be either x + 1 x+1 or y + 1 y+1 , and not x + 2 x+2 . Given both sides satisfy the same condition, the equation shall be TRUE (it shall be equal sign, not \neq . The question said it is \neq , hence the answer is FALSE )

Both sides satisfy the same condition. Given statement says both sides are not equal .

Therefore given statement is FALSE .

Zds Alpha - 5 years, 4 months ago

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Yes, I illustrated in the solution that both sides are equal. If it say "not equal", then the statement is FALSE .

Kay Xspre - 5 years, 4 months ago
Zds Alpha
Feb 2, 2016

Lets take left hand side:

L.H.S = (x∘y=x+1) OR (x∘y=y+1)

According to rules: x∘y = x+1 if x>y And x∘y = y+1 if y>x We can write,

L.H.S = x>y OR y>x

Either x is greater or y. Thus we can say x≠y

L.H.S = x≠y

Lets take right hand side:

R.H.S = (x∘y≠x+2)

According to first rules: x∘y=x+2=y+2 if x=y (In other words x∘x=x+2)

We can also write this as: x∘y≠x+2≠y+2 if x≠y

R.H.S = x≠y

Left hand side = Right hand side

((x∘y=x+1) OR (x∘y=y+1))=(x∘y≠x+2) Is True

But ((x∘y=x+1) OR (x∘y=y+1))≠(x∘y≠x+2) Is False

Useful Information:

The operator ∘ is called Zeration also known as Hyper0,Increment and Successor.

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