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1) \(xyz - xy = 0\) 2) Either \(x=0,\) or \(y=0\) or \(z=1\)

The see the OA, I think it's incorrect. What's your opinion?

Dear suk1234, I'm happy to respond.

I think the OA of (E) is correct. First of all, Statement #1 and Statement #2 are logical equivalent, and one implies the other. If x = 0 or y = 0, then the other could equal anything, z could equal anything, and the product would be zero. If z = 1, then x & y could equal anything, and the product could equal any value on the continuous infinity of the number line. Thus, there's no way to determine a definitive value of the product.

Does this make sense? Please let me know if you have any further questions. Mike _________________

1) \(xyz - xy = 0\) 2) Either \(x=0,\) or \(y=0\) or \(z=1\)

The see the OA, I think it's incorrect. What's your opinion?

Dear suk1234, I'm happy to respond.

I think the OA of (E) is correct. First of all, Statement #1 and Statement #2 are logical equivalent, and one implies the other. If x = 0 or y = 0, then the other could equal anything, z could equal anything, and the product would be zero. If z = 1, then x & y could equal anything, and the product could equal any value on the continuous infinity of the number line. Thus, there's no way to determine a definitive value of the product.

Does this make sense? Please let me know if you have any further questions. Mike

Thank you Mike for the Quick rescue!

Here is how I evaluated Statement 1:

Z=1 and Either X=0 or Y=0 or Both X and Y = 0

Then evaluate all the possible values! 1. (X=0) 0*Y*1=0 2. (Y=0) X*0*1=0 3. (X and Y = 0) 0*0*1=0

I think in case of statement 2 this reasoning doesn't apply because it presents three cases which may or may not be true ( \(x=0,\) or \(y=0\) or \(z=1\) either of these can happen or not). But in case of statement 1 we are definitely sure about the value of XY.

Then evaluate all the possible values! 1. (X=0) 0*Y*1=0 2. (Y=0) X*0*1=0 3. (X and Y = 0) 0*0*1=0

I think in case of statement 2 this reasoning doesn't apply because it presents three cases which may or may not be true ( \(x=0,\) or \(y=0\) or \(z=1\) either of these can happen or not). But in case of statement 1 we are definitely sure about the value of XY.

Dear suk1234 I'm happy to respond.

Statement #1 says xyz - xy = 0 Add xy to both sides: xyz = xy (xy)*z = (xy)

Here, we are presented with a choice. Case One: If (xy) does not equal zero, then we can divide by (xy), and get z = 1. That's one case, in which (xy) can have any value on the number line other than zero, and z = 1. Here, the product xyz would be equal to xy, and could be anything other than zero. Case Two: If (xy) = 0, then z could be anything on the number line. This is the other case. If (xy) = 0, then either x = 0 or y = 0, which will make the product equal zero. (Here, z could be 1, or it could be anything else on the number line.)

You see, the crucial mathematical word is the word "or" ---- either z = 1 OR (xy) = 0. You are interpreting the two requirements as if they are simultaneous, not a mutually exclusive choice. The two cases are actually mutually exclusive.

Then evaluate all the possible values! 1. (X=0) 0*Y*1=0 2. (Y=0) X*0*1=0 3. (X and Y = 0) 0*0*1=0

I think in case of statement 2 this reasoning doesn't apply because it presents three cases which may or may not be true ( \(x=0,\) or \(y=0\) or \(z=1\) either of these can happen or not). But in case of statement 1 we are definitely sure about the value of XY.

Dear suk1234 I'm happy to respond.

Statement #1 says xyz - xy = 0 Add xy to both sides: xyz = xy (xy)*z = (xy)

Here, we are presented with a choice. Case One: If (xy) does not equal zero, then we can divide by (xy), and get z = 1. That's one case, in which (xy) can have any value on the number line other than zero, and z = 1. Here, the product xyz would be equal to xy, and could be anything other than zero. Case Two: If (xy) = 0, then z could be anything on the number line. This is the other case. If (xy) = 0, then either x = 0 or y = 0, which will make the product equal zero. (Here, z could be 1, or it could be anything else on the number line.)

You see, the crucial mathematical word is the word "or" ---- either z = 1 OR (xy) = 0. You are interpreting the two requirements as if they are simultaneous, not a mutually exclusive choice. The two cases are actually mutually exclusive.

Does all this make sense? Mike

Oh I see, where I was going wrong with it.Thank Mike that was an amazing explanation.

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