GATE CSE 2015 Set 3 | Set Theory & Algebra Question 20 | Discrete Mathematics

GATE CSE 2015 Set 3

MCQ (Single Correct Answer)

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Suppose $$𝑈$$ is the power set of the set $$S = \left\{ {1,2,3,4,5,6,} \right\}$$. For any $$T \in U,$$ let $$\left| T \right|$$ denote the number of elements in $$𝑇$$ and $$T'$$ denote the complement of $$𝑇.$$ For any $$T,R \in U,$$ let $$T\backslash R$$ be the set of all elements in $$𝑇$$ which are not in $$𝑅.$$ Which one of the following is true?

$$\forall X \in U\,\,$$ $$\left( {\left| X \right| = \left| {X'} \right|} \right)$$

$$\exists X \in U$$ $$\exists Y \in U\,\,$$ $$\left( {\left| X \right| = 5,\left| Y \right| = 5} \right.$$ and $$\left. {X \cap Y = \phi } \right)$$

$$\forall X \in U\,$$ $$\forall Y \in U\,\,$$ $$\,\,\left( {\left| X \right| = 2,\left| Y \right| = 3{\mkern 1mu} {\mkern 1mu} {\mkern 1mu} {\mkern 1mu} and{\mkern 1mu} {\mkern 1mu} {\mkern 1mu} {\mkern 1mu} X\backslash Y = \phi } \right)$$

$$\forall X \in U\,\,$$ $$\forall Y \in U\,\,$$ $$\,\left( {X\backslash Y = Y'\backslash X'} \right)$$

GATE CSE 2015 Set 2

Numerical

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The cardinally of the power set of $$\left\{ {0,1,2,\,\,....,\,\,10} \right.\left. \, \right\}$$ is _____________.

Your input ____

GATE CSE 2015 Set 2

MCQ (Single Correct Answer)

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Let $$𝑅$$ be the relation on the set of positive integers such that $$aRb$$ if and only if $$𝑎 $$ and $$𝑏$$ are distinct and have a common divisor other than $$1.$$ Which one of the following statements about $$𝑅$$ is true?

$$𝑅$$ is symmetric and reflexive but not transitive

$$𝑅$$ is reflexive but not symmetric and not transitive

$$𝑅$$ is transitive but not reflexive and not symmetric

$$𝑅$$ is symmetric but not reflexive and not transitive

GATE CSE 2014 Set 3

MCQ (Single Correct Answer)

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Let $$X$$ and $$Y$$ be finite sets and $$f:X \to Y$$ be a function. Which one of the following statements is TRUE?

For any subsets $$A$$ and $$B$$ of $$X$$, $$\left| {f\left( {A \cup B} \right)} \right| = \left| {f\left( A \right)} \right| + \left| {f\left( B \right)} \right|$$

For any subsets $$A$$ and $$B$$ of $$X$$, $${f\left( {A \cap B} \right)}$$ $$=$$ $$f\left( A \right) \cap f\left( B \right)$$

For any subsets $$\left| {f\left( {A \cap B} \right)} \right| = \min \left\{ {\left| {f\left( A \right)} \right|,\left| {f\left( B \right)} \right|} \right\}$$

for any subsets $$S$$ and $$T$$ of $$Y$$, $${f^{ - 1}}\left( {S \cap T} \right) = {f^{ - 1}}\left( S \right) \cap {f^{ - 1}}\left( T \right)$$

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