Question

We want to determine whether or not for all A, B, C ⊆ U, A ∩ (B ∩ C)' = (A - B) U (A - C) is an identity. Which one of the following alternatives is the correct way to determine this? Select one: a. x ∈ A ∩ (B ∩ C)' iff x ∈ A and x ∈ (B ∩ C)' iff x ∈ A and x ∈ (B ∩ C) iff x ∈ A and (x ∉ B or x ∉ C) iff (x ∈ A and x ∉ B) or (x ∈ A and x ∉ C) iff x ∈ (A - B) or x ∈ (A - C) iff x ∈ (A - B) U (A - C), which is equal to the RHS of the given equation. We can therefore say that the given statement is an identity. b. x ∈ A ∩ (B ∩ C)' iff x ∈ A and x ∈ (B ∩ C)' iff x ∈ A and x ∈ (B - C) iff x ∈ A and (x ∉ B or x ∉ C) iff (x ∈ A and x ∉ B) or (x ∈ A and x ∉ C) iff x ∈ (A - B) or x ∈ (A - C) iff x ∈ (A - B) U (A - C), which is equal to the RHS of the given equation. We can therefore say that the given statement is an identity.

          We want to determine whether or
not for all A, B, C ⊆ U,
A ∩ (B ∩ C)' = (A - B) U (A -
C) is an identity.
Which one of the following
alternatives is the correct way to
determine this?
Select one:
a.
x ∈ A ∩ (B ∩ C)'
iff x ∈ A and x ∈ (B ∩ C)'
iff x ∈ A and x ∈ (B ∩ C)
iff x ∈ A and (x ∉ B or x ∉ C)
iff (x ∈ A and x ∉ B) or (x ∈ A and x ∉ C)
iff x ∈ (A - B) or x ∈ (A - C)
iff x ∈ (A - B) U (A - C), which is equal to the RHS of the given equation.
We can therefore say that the given statement is an identity.
b.
x ∈ A ∩ (B ∩ C)'
iff x ∈ A and x ∈ (B ∩ C)'
iff x ∈ A and x ∈ (B - C)
iff x ∈ A and (x ∉ B or x ∉ C)
iff (x ∈ A and x ∉ B) or (x ∈ A and x ∉ C)
iff x ∈ (A - B) or x ∈ (A - C)
iff x ∈ (A - B) U (A - C), which is equal to the RHS of the given equation.
We can therefore say that the given statement is an identity.

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we want to determine whether or not for all a b c u a b c a b u a c is an identity which one of the following alternatives is the correct way to determine this select one a x a b c iff x a a 46215

Added by Pilar C.

Calculus: Early Transcendentals

James Stewart 8th Edition

Instant Answer

Solved by Expert Sri K

04/10/2023

Step 1

Step 1: To determine whether the given statement is an identity, we need to show that the left-hand side (LHS) is equal to the right-hand side (RHS) for all possible sets A, B, and C. Show more…

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We want to determine whether or not for all A, B, C ⊆ U, A ∩ (B ∩ C)' = (A - B) U (A - C) is an identity. Which one of the following alternatives is the correct way to determine this? Select one: a. x ∈ A ∩ (B ∩ C)' iff x ∈ A and x ∈ (B ∩ C)' iff x ∈ A and x ∈ (B ∩ C) iff x ∈ A and (x ∉ B or x ∉ C) iff (x ∈ A and x ∉ B) or (x ∈ A and x ∉ C) iff x ∈ (A - B) or x ∈ (A - C) iff x ∈ (A - B) U (A - C), which is equal to the RHS of the given equation. We can therefore say that the given statement is an identity. b. x ∈ A ∩ (B ∩ C)' iff x ∈ A and x ∈ (B ∩ C)' iff x ∈ A and x ∈ (B - C) iff x ∈ A and (x ∉ B or x ∉ C) iff (x ∈ A and x ∉ B) or (x ∈ A and x ∉ C) iff x ∈ (A - B) or x ∈ (A - C) iff x ∈ (A - B) U (A - C), which is equal to the RHS of the given equation. We can therefore say that the given statement is an identity.

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Transcript

00:01 First we have to, hello everyone let us look into the equation.

00:04 Here first we have to show that 1 implies 2.

00:07 So, now, first consider a intersection b equal to 5.

00:12 Suppose if x belongs to the set a, then x does not belongs to the set b.

00:18 So, which means that x belongs to b complement.

00:23 From this we can say that x belongs to a and x belongs to b complement from this a is a subset of b complement.

00:31 Then next to consider 2 implies 1 that is the second statement implies the first statement.

00:38 So, here we can have a is a subset of b complement.

00:42 So, if x belongs to a, then we can have x belongs to b complement.

00:48 So, now, x belongs to b complement.

00:50 So, which implies x does not belongs to b.

00:53 Hence, we can say that x does not belongs to a intersection b.

00:58 Therefore, we can say that a intersection b is equal to null set.

01:02 So, by using this condition we can prove both the statements.

01:06 Next one 2 implies 3.

01:09 So, here it is given a is a subset of b complement.

01:14 So, which implies a intersection b equal to null set.

01:18 Hence, we can have for every x in b, x does not belongs to a, we can have x does not belongs to a complement.

01:34 X does not belongs to a implies x belongs to a complement...

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