B.TECH - Semester 3 discrete structures Question Paper 2020 (apr)
Practice authentic previous year university questions for better exam preparation.
- (i) $p \rightarrow(q \vee r) \leftrightarrow(p \rightarrow q) v(p \rightarrow r)$ (ii) $\neg(p \rightarrow q) \leftrightarrow p \wedge \neg q$
- (b) Let set $A=\{1,2,3,4\}$ and let $R$ be an equivalence relation on $A$ partitions $A$ into $\{1,2\},\{3\},\{4)$. Find the relation $R$.
- (a) Give a relation which is both symmetric and antisymmetric on set $A=\{1,2,3,4\}$.
- (d) Let set $A=\{1,2,3,4\}$ and $R$ is a relation on $A$ defined by $R=\{(1,1),(2,2),(1,2),(2,1)$. Find the properties of the relation. Is it an equivalence relation? OR
- (b) Let $N$ be the set of natural numbers and $f$ is a function from $N$ to $N$ defined by $f(x)=3 x+7$. Check whether the function is bijective.
- (a) Let G be the set of rational numbers excluding 1 and define *on $G$ as $a{ }^{*} b=a+b-a b$ for all $a, b \in c$. Show that ( $G,{ }^{*}$ ) is a group. Check whether it is an abelian group. 10
- (c) Prove by mathematical induction $1+2+3+\ldots+n=n(n+1) / 2$.
- (a) Symbolize the following statement using predicate logic "All world loves a lover"
- (b) Verify the validity of the following argument." Every living thing is a planet or an animal. John's gold fish is alive and is not a planet. All animals have hearts. Therefore John's gold fish has a heart".
- (c) Let set $A=\{1,2,3,4\}$, draw the graph representing the relation $R$ on $A$ defined by $R=(1,2),(1,1),(23)(3,2),(3,3),(4,3),(4,1)\}$.
- (a) Let the sets $A$ and $B$ defined as $A=\{1,2)$ and $B=\{7,8,9\}$ and a relation $f$ is defined from $A$ to $B$ as $f=\{(1,7),(2,8),(1,9)\}$. Is $f$ a function? Justify your answer.
- (b) Prove that intersection of subgroups of a group $G$ is a subgroup of $G$. 10
- (a) State and prove Lagranges Theorem on subgroups 10
- (b) Define a ring with example 10
- Draw the Hasse diagram for the partial ordering "divides" on the set $D_{100}=\{1,2,4,5,10,20,25,50,100\}$. Check whether it is a lattice. Justify. If it is a lattice find its sub lattices.
- (a) Draw a complete graph with $2,3,4$, and 5 verices.
- (b) What are the various representation of graphs? Explain with examples
- Write the equivalent form of the propositional logic formula using $\neg$ and $v$ only $p \rightarrow(q \rightarrow r)$
- Prove that $p \vee \neg p$ is a tautology
- Define partially ordered set and equivalence relation with example
- If $A$ and $B$ are two sets, prove that $A \cup B=B \cup A$
- If $A$ and $B$ are two sets, prove that $A-B \neq B-A$.
- Given a set $A=\{1,2,3,4\}$ and a relation $R$ on $A$ is defined as $R=[(1,2),(2,3),(3,4),(4,1)\}$. Find the inverse of the relation $R$
- Define abelian group with example
- Define (a)Integral domain (b) Field
- What is a connected component of a graph. Draw a graph with 2 connected components
- Define a binary tree. Draw a complete binary tree with 7 vertices Answer any one full questions from each Module. Each question carries 20 marks.
- (a) Write the truth table for the statement by representing in propositional logic, " If a quadrilateral $A B C D$ is a square then all the sides of it are equal". Also write the converse, inverse and contra positive of the given statement with truth table. Which one of the above is equivalent to th...
- (b) Show that
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