If X and Y are two sets, then X ∩ (X ∪...

MathematicsTypes of Sets and Set OperationsJEE Advanced 1979Easy

If $X$ and $Y$ are two sets, then $X \cap (X \cup Y)^{c}$ equals.

(A)

X

(B)

Y

(C)

ϕ

(D)

None of these

Answer:C

Visualized Solution (English)

Visualizing the Universal Set

Let $U$ be the Universal set.
Let $X$ and $Y$ be two subsets of $U$ .
The circles represent the elements belonging to each set.

Defining the Union .math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } X \cup Y

The union $X \cup Y$ consists of all elements in $X$ , $Y$ , or both.
Visually, it is the total area covered by both circles.

Finding the Complement .math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } (X \cup Y)^{c}

The complement $(X \cup Y)^{c}$ contains elements in $U$ that are NOT in $X \cup Y$ .
By De Morgan's Law: $(X \cup Y)^{c} = X^{c} \cap Y^{c}$ .

The Intersection .math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } X \cap (X \cup Y)^{c}

We need to find the common elements between $X$ and $(X \cup Y)^{c}$ .
Since $(X \cup Y)^{c}$ is strictly outside $X$ , there are no common elements.

Final Result: The Null Set .math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } ϕ

$X \cap (X \cup Y)^{c} = X \cap (X^{c} \cap Y^{c})$
Using associative law: $(X \cap X^{c}) \cap Y^{c}$
Since $X \cap X^{c} = ϕ$ , we get $ϕ \cap Y^{c} = ϕ$ .
The correct option is (2).

00:00 / 00:00

Conceptually Similar Problems

MathematicsTypes of Sets and Set OperationsJEE Main 2009Easy

View in:English Hindi

If A, B and C are three sets such that $A \cap B = A \cap C$ and $A \cup B = A \cup C$ , then

(A)

A=C

(B)

B=C

(C)

A \cap B = \phi

(D)

A=B

Answer:B

MathematicsTypes of Sets and Set OperationsJEE Main 2014Easy

View in:English Hindi

If $X = {4^{n} - 3 n - 1 : n \in N}$ and $Y = {9 (n - 1) : n \in N}$ , where N is the set of natural numbers, then $X \cup Y$ is equal to:

(A)

(B)

(C)

(D)

Y - X

Answer:B

MathematicsAddition and Multiplication TheoremsJEE Advanced 1985Easy

View in:English Hindi

$P (A \cup B) = P (A \cap B)$ if and only if the relation between $P (A)$ and $P (B)$ is .........

Answer:P(A) = P(B)

MathematicsAddition and Multiplication TheoremsJEE Advanced 1988Easy

View in:English Hindi

For two given events $A$ and $B, P (A \cap B)$ is

* Multiple Correct Options

(A)

not less than

P (A) + P (B) - 1

(B)

not greater than

P (A) + P (B)

(C)

equal to

P (A) + P (B) - P (A \cup B)

(D)

equal to

P (A) + P (B) + P (A \cup B)

Answer:A, B, C

MathematicsTypes of Sets and Set OperationsJEE Advanced 1980Easy

View in:English Hindi

(i) Set $A$ has 3 elements, and set $B$ has 6 elements. What can be the minimum number of elements in the set $A \cup B$ ? (ii) $P, Q, R$ are subsets of a set $A$ . Is the following equality true? $R \times (P^{c} \cup Q^{c})^{c} = (R \times P) \cap (R \times Q)$ ?

Answer:(i) 6 (ii) Yes

MathematicsConditional ProbabilityJEE Advanced 1982Easy

View in:English Hindi

If $A$ and $B$ are two events such that $P (A) > 0$ , and $P (B) \neq = 1$ , then $P (\frac{A ˉ}{B ˉ})$ is equal to

(A)

1 - P (\frac{A}{B})

(B)

1 - P (\frac{A ˉ}{B})

(C)

\frac{1 - P ( A \cup B )}{P ( B ˉ )}

(D)

\frac{P ( A ˉ )}{P ( B ˉ )}

Answer:C

MathematicsAddition and Multiplication TheoremsJEE Advanced 1991Easy

View in:English Hindi

For any two events $A$ and $B$ in a sample space

* Multiple Correct Options

(A)

P (A / B) \geq \frac{P ( A ) + P ( B ) - 1}{P ( B )}, P (B) \neq = 0

is always true

(B)

P (A \cap \overset{ˉ}{B}) = P (A) - P (A \cap B)

does not hold

(C)

P (A \cup B) = 1 - P (\overset{ˉ}{A}) P (\overset{ˉ}{B})

, if

A

and

B

are independent

(D)

P (A \cup B) = 1 - P (\overset{ˉ}{A}) P (\overset{ˉ}{B})

, if

A

and

B

are disjoint.

Answer:A, C

MathematicsTechniques of DifferentiationJEE Main 2006Easy

View in:English Hindi

If $x^{m} \cdot y^{n} = (x + y)^{m + n}$ , then $\frac{d y}{d x}$ is

(A)

\frac{y}{x}

(B)

\frac{x + y}{x y}

(C)

x y

(D)

\frac{x}{y}

Answer:A

MathematicsAddition and Multiplication TheoremsJEE Advanced 1995Easy

View in:English Hindi

Let $0 < P (A) < 1, 0 < P (B) < 1$ and $P (A \cup B) = P (A) + P (B) - P (A) P (B)$ then

* Multiple Correct Options

(A)

P (B / A) = P (B) - P (A)

(B)

P (A^{c} - B^{c}) = P (A^{c}) - P (B^{c})

(C)

P (A \cup B)^{c} = P (A^{c}) P (B^{c})

(D)

P (A / B) = P (A)

Answer:C, D

MathematicsFundamental Principle of CountingJEE Main 2012Easy

View in:English Hindi

Let $X = {1, 2, 3, 4, 5}$ . The number of different ordered pairs (Y,Z) that can formed such that $Y \subseteq X, Z \subseteq X$ and $Y \cap Z$ is empty is:

(A)

5²

(B)

3⁵

(C)

2⁵

(D)

5³

Answer:B

If $X$ and $Y$ are two sets, then $X \cap (X \cup Y)^{c}$ equals.

Visualized Solution (English)

Visualizing the Universal Set

Defining the Union $X \cup Y$

Finding the Complement $(X \cup Y)^{c}$

The Intersection $X \cap (X \cup Y)^{c}$

Final Result: The Null Set $ϕ$

Conceptually Similar Problems

If A, B and C are three sets such that $A \cap B = A \cap C$ and $A \cup B = A \cup C$ , then

If $X = {4^{n} - 3 n - 1 : n \in N}$ and $Y = {9 (n - 1) : n \in N}$ , where N is the set of natural numbers, then $X \cup Y$ is equal to:

$P (A \cup B) = P (A \cap B)$ if and only if the relation between $P (A)$ and $P (B)$ is .........

For two given events $A$ and $B, P (A \cap B)$ is

(i) Set $A$ has 3 elements, and set $B$ has 6 elements. What can be the minimum number of elements in the set $A \cup B$ ? (ii) $P, Q, R$ are subsets of a set $A$ . Is the following equality true? $R \times (P^{c} \cup Q^{c})^{c} = (R \times P) \cap (R \times Q)$ ?

If $A$ and $B$ are two events such that $P (A) > 0$ , and $P (B) \neq = 1$ , then $P (\frac{A ˉ}{B ˉ})$ is equal to

For any two events $A$ and $B$ in a sample space

If $x^{m} \cdot y^{n} = (x + y)^{m + n}$ , then $\frac{d y}{d x}$ is

Let $0 < P (A) < 1, 0 < P (B) < 1$ and $P (A \cup B) = P (A) + P (B) - P (A) P (B)$ then

Let $X = {1, 2, 3, 4, 5}$ . The number of different ordered pairs (Y,Z) that can formed such that $Y \subseteq X, Z \subseteq X$ and $Y \cap Z$ is empty is:

If $X$ and $Y$ are two sets, then $X \cap (X \cup Y)^{c}$ equals.

If A, B and C are three sets such that $A \cap B = A \cap C$ and $A \cup B = A \cup C$ , then

If $X = {4^{n} - 3 n - 1 : n \in N}$ and $Y = {9 (n - 1) : n \in N}$ , where N is the set of natural numbers, then $X \cup Y$ is equal to:

$P (A \cup B) = P (A \cap B)$ if and only if the relation between $P (A)$ and $P (B)$ is .........

For two given events $A$ and $B, P (A \cap B)$ is

(i) Set $A$ has 3 elements, and set $B$ has 6 elements. What can be the minimum number of elements in the set $A \cup B$ ? (ii) $P, Q, R$ are subsets of a set $A$ . Is the following equality true? $R \times (P^{c} \cup Q^{c})^{c} = (R \times P) \cap (R \times Q)$ ?

If $A$ and $B$ are two events such that $P (A) > 0$ , and $P (B) \neq = 1$ , then $P (\frac{A ˉ}{B ˉ})$ is equal to

For any two events $A$ and $B$ in a sample space

If $x^{m} \cdot y^{n} = (x + y)^{m + n}$ , then $\frac{d y}{d x}$ is

Let $0 < P (A) < 1, 0 < P (B) < 1$ and $P (A \cup B) = P (A) + P (B) - P (A) P (B)$ then

Let $X = {1, 2, 3, 4, 5}$ . The number of different ordered pairs (Y,Z) that can formed such that $Y \subseteq X, Z \subseteq X$ and $Y \cap Z$ is empty is:

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } If X and Y are two sets, then X∩(X∪Y)c equals.

Visualized Solution (English)

Visualizing the Universal Set

Defining the Union .math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } X∪Y

Finding the Complement .math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } (X∪Y)c

The Intersection .math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } X∩(X∪Y)c

Final Result: The Null Set .math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } ϕ

Conceptually Similar Problems

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } If A, B and C are three sets such that A∩B=A∩C and A∪B=A∪C, then

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } If X={4n−3n−1:n∈N} and Y={9(n−1):n∈N}, where N is the set of natural numbers, then X∪Y is equal to:

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } P(A∪B)=P(A∩B) if and only if the relation between P(A) and P(B) is .........

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } For two given events A and B,P(A∩B) is

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.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } For any two events A and B in a sample space

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } If xm⋅yn=(x+y)m+n, then dxdy​ is

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } Let 0<P(A)<1,0<P(B)<1 and P(A∪B)=P(A)+P(B)−P(A)P(B) then

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } Let X={1,2,3,4,5}. The number of different ordered pairs (Y,Z) that can formed such that Y⊆X,Z⊆X and Y∩Z is empty is:

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } If X and Y are two sets, then X∩(X∪Y)c equals.

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } If A, B and C are three sets such that A∩B=A∩C and A∪B=A∪C, then

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.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } P(A∪B)=P(A∩B) if and only if the relation between P(A) and P(B) is .........

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.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } For any two events A and B in a sample space

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } If xm⋅yn=(x+y)m+n, then dxdy​ is

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.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } Let X={1,2,3,4,5}. The number of different ordered pairs (Y,Z) that can formed such that Y⊆X,Z⊆X and Y∩Z is empty is:

If $X$ and $Y$ are two sets, then $X \cap (X \cup Y)^{c}$ equals.

Defining the Union $X \cup Y$

Finding the Complement $(X \cup Y)^{c}$

The Intersection $X \cap (X \cup Y)^{c}$

Final Result: The Null Set $ϕ$

If A, B and C are three sets such that $A \cap B = A \cap C$ and $A \cup B = A \cup C$ , then

If $X = {4^{n} - 3 n - 1 : n \in N}$ and $Y = {9 (n - 1) : n \in N}$ , where N is the set of natural numbers, then $X \cup Y$ is equal to:

$P (A \cup B) = P (A \cap B)$ if and only if the relation between $P (A)$ and $P (B)$ is .........

For two given events $A$ and $B, P (A \cap B)$ is

If $A$ and $B$ are two events such that $P (A) > 0$ , and $P (B) \neq = 1$ , then $P (\frac{A ˉ}{B ˉ})$ is equal to

For any two events $A$ and $B$ in a sample space

If $x^{m} \cdot y^{n} = (x + y)^{m + n}$ , then $\frac{d y}{d x}$ is

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If $X = {4^{n} - 3 n - 1 : n \in N}$ and $Y = {9 (n - 1) : n \in N}$ , where N is the set of natural numbers, then $X \cup Y$ is equal to:

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