Sets

Probability Level 1

True or False?

∣ X C ∪ Y C ∪ Z C ∣ = ∣ ( X ∩ Y ∩ Z ) C ∣ \left|X^C \cup Y^C \cup Z^C\right| = \left|(X \cap Y \cap Z)^C\right|

True False There isn't enough information

Zandra Vinegar by Zandra Vinegar

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3 solutions

Eli Ross Staff
Apr 18, 2016

The easiest way to deal with this is to think about the sets each side is counting, in words.

The set in the left-hand-side is all things that are not in X, or not in Y, or not in Z. In other words, it's all the things that are not in all 3 of X, Y, and Z.

But wait! "All the things things not in all 3 of X, Y, and Z" is exactly the set that is on the right-hand-side; specifically, the complement of the intersection of X, Y, and Z.

20 Helpful 3 Interesting 0 Brilliant 0 Confused

Is this a de morgan's law extension?

rajdeep das - 4 years, 10 months ago

yeah nice method or one can deal with it just by solving the problems and talking a U set to , but you got it right to :) thanks man.

Abhishek Kumar - 3 years, 2 months ago

Hmmm...asks for the measure of the sets, not what's in them. Does't that negate the answer?

Joe Horton - 3 years ago
Bruno Martel
Apr 1, 2021

I thought it as everything excepts the intersection of the 3 sets

0 Helpful 0 Interesting 0 Brilliant 0 Confused
Soham Nimale
May 22, 2020

Draw three circles intercepting each other and see if the statement is true......ya it is

And

X^c = complementary of X set

0 Helpful 0 Interesting 0 Brilliant 0 Confused

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