Question

Consider the universal set U and the sets X, Y, Z. U={1,2,3,4,5,6} X={2,5,6} Y={4,5,6} Z={2,3,4} What is (Y⋂X′)⋃Z′?

Answer 1

The set (Y⋂X′)⋃Z′ is {1, 4, 5, 6}.

Step-by-step explanation:

Here's the step-by-step calculation:

X' (complement of X):

X' contains all elements in the universal set U that are not in X.

X' = U - X = {1, 3}

Y⋂X' (intersection of Y and X'):

This set contains elements that are common to both Y and X'.

Y⋂X' = {4}

Z' (complement of Z):

Z' contains all elements in U that are not in Z.

Z' = U - Z = {1, 5, 6}

(Y⋂X')⋃Z' (union of Y⋂X' and Z'):

This set combines all elements from both Y⋂X' and Z'.

(Y⋂X')⋃Z' = {1, 4} ⋃ {1, 5, 6} = {1, 4, 5, 6}

Therefore, the final answer is {1, 4, 5, 6}.

Answer 2

To solve (Y⋂X′)⋃Z′, one must find the complements X′ and Z′ within the universal set, and calculate the intersection and unions required. The final result is the set {1,4,5,6}.

Step-by-step explanation:

The problem at hand deals with set theory operations within a universal set U. To find the set (Y⋂X′)⋃Z′, where ' means the complement in U, we perform the following steps:

• Find X′, the complement of X in U: X′ = {1,3,4}.
• Calculate the intersection of Y and X′: Y⋂X′ = {4}.
• Find Z′, the complement of Z in U: Z′ = {1,5,6}.
• Finally, unite the sets Y⋂X′ and Z′: (Y⋂X′)⋃Z′ = {1,4,5,6}.