Let A, B and C be three sets. Then A B - C is not equal to Option 1: ((A - C)^c (B - C)^c)^c Option 2: (A^c C) (B^c C) Option 3: (A - C) (B - C) Option 4: ((A^c C) (B^c C))^c

Answer: (A^c C) (B^c C). Solution: We need to find which option is not equal to A B - C. First, let's understand what A B - C means. It represents all elements that are in A or B, but not in C. In set notation: A B - C = (A B) C^c Now let's check each option: Checking Option 3: (A - C) (B - C) (A - C) means elements in A but not in C (B - C) means elements in B but not in C So (A - C) (B - C) means elements that are in A or B, but not in C This is exactly A B - C ✓ Checking Option 1: ((A - C)^c (B - C)^c)^c Let's work from inside out: (A - C)^c = (A C^c)^c = A^c C (using De Morgan's law) (B - C)^c = (B C^c)^c = B^c C Now: (A - C)^c (B - C)^c = (A^c C) (B^c C) Taking complement: ((A^c C) (B^c C))^c = (A^c C)^c (B^c C)^c = (A C^c) (B C^c) = (A - C) (B - C) = A B - C ✓ Checking Option 4: ((A^c C) (B^c C))^c This is the same expression as in Option 1, which equals A B - C ✓ Checking Option 2: (A^c C) (B^c C) Simplifying: A^c C B^c C = A^c B^c C This means elements that are not in A, or not in B, or in C. This is very different from A B - C (which excludes elements in C). For example, if an element is in C, it will be in (A^c B^c C) but not in A B - C. Therefore, Option 2 is not equal to A B - C ✗

Modern Math

Set Theory

Medium

Let A, B and C be three sets. Then $A \cup B - C$ is not equal to

$((A - C)^c \cap (B - C)^c)^c$

$(A^c \cup C) \cup (B^c \cup C)$

$(A - C) \cup (B - C)$

$((A^c \cup C) \cap (B^c \cup C))^c$

✅ Correct Option: 2

We need to find which option is not equal to $A \cup B - C$ .

First, let's understand what $A \cup B - C$ means. It represents all elements that are in $A$ or $B$ , but not in $C$ .

In set notation: $A \cup B - C = (A \cup B) \cap C^c$

Now let's check each option:

Checking Option 3: $(A - C) \cup (B - C)$

$(A - C)$ means elements in $A$ but not in $C$

$(B - C)$ means elements in $B$ but not in $C$

So $(A - C) \cup (B - C)$ means elements that are in $A$ or $B$ , but not in $C$

This is exactly $A \cup B - C$ ✓

Checking Option 1: $((A - C)^c \cap (B - C)^c)^c$

Let's work from inside out:

$(A - C)^c = (A \cap C^c)^c = A^c \cup C$ (using De Morgan's law)

$(B - C)^c = (B \cap C^c)^c = B^c \cup C$

Now: $(A - C)^c \cap (B - C)^c = (A^c \cup C) \cap (B^c \cup C)$

Taking complement: $((A^c \cup C) \cap (B^c \cup C))^c = (A^c \cup C)^c \cup (B^c \cup C)^c$

$= (A \cap C^c) \cup (B \cap C^c) = (A - C) \cup (B - C) = A \cup B - C$ ✓

Checking Option 4: $((A^c \cup C) \cap (B^c \cup C))^c$

This is the same expression as in Option 1, which equals $A \cup B - C$ ✓

Checking Option 2: $(A^c \cup C) \cup (B^c \cup C)$

Simplifying: $A^c \cup C \cup B^c \cup C = A^c \cup B^c \cup C$

This means elements that are not in $A$ , or not in $B$ , or in $C$ .

This is very different from $A \cup B - C$ (which excludes elements in $C$ ).

For example, if an element is in $C$ , it will be in $(A^c \cup B^c \cup C)$ but not in $A \cup B - C$ .

Therefore, Option 2 is not equal to $A \cup B - C$ ✗

2025: Sample Paper

Let $A \times B$ denote the Cartesian product of sets $A$ and $B$ . If the cardinality of $A \cap B$ is $10$ , then the cardinality of $(A \times B) \cap (B \times A)$ is:

2025: Sample Paper

Which of the following relations is an equivalence relation?

2025: Sample Paper

Let A, B, and C be three sets. Then, A-B-C is not equal to

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