(B) We evaluate the expression $(A \setminus B) \setminus C$: By definition of set difference, $A \setminus B = A \cap B'$. Thus, $(A \setminus B) \se

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$(A \setminus B) \setminus C = A \setminus (B \cup C)$

(B) We evaluate the expression $(A \setminus B) \setminus C$: By definition of set difference, $A \setminus B = A \cap B'$. Thus, $(A \setminus B) \setminus C = (A \cap B') \cap C'$. Using the associative property of intersection, we get $A \cap (B' \cap C')$. By De Morgan's Law, $B' \cap C' = (B \cup C)'$. Therefore, $(A \setminus B) \setminus C = A \cap (B \cup C)' = A \setminus (B \cup C)$. Hence, the statement in option $B$ is valid.

Class 11 Mathematics Set Theory Basic of Set theory

Easy

Let $A, B, C$ be subsets of a set $X$. Which of the following set-theoretic statements is valid?

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A
$A \cup (B \setminus C) = (A \cup B) \setminus (A \cup C)$
B
$(A \setminus B) \setminus C = A \setminus (B \cup C)$
C
$(A \cup B) \setminus A = A \setminus B$
D
$A \setminus C = B \setminus C$

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Let $A, B, C$ be subsets of a set $X$. Which of the following set-theoretic statements is valid?

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List all the elements of the following set:$F = \{ x : x \text{ is a consonant in the English alphabet which precedes } k \}$

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List all the elements of the following set:
$F = \{ x : x \text{ is a consonant in the English alphabet which precedes } k \}$

Examine whether the following statement is true or false:
$\{x : x \text{ is an even natural number less than } 6\} \subset \{x : x \text{ is a natural number which divides } 36\}$

In each of the following,determine whether the statement is true or false. If it is true,prove it. If it is false,give a counterexample.
If $x \in A$ and $A \not\subset B$,then $x \in B$