Basic Mathematics for IT — Winter 2018

Give definition of the following terms: 1) Null set 2) Union of two sets 3) Symmetric Matrix 4) Universal Quantifiers 5) Reflexive Relation 6) Mixed Graph 7) Isolated vertex

Let U = {a,b,c,d,e,f,g,h,p,q,r}X = {a,b,c,d,e}Y = {c,d,e,f,g,h}Z = {h,p,q,r}Compute, XY, YZX, X-(YZ), (XY)’, X’, X’Y’, XΔY, using Venn Diagram. (Note: XΔY = (X- Y)(Y-X)

Check whether the statements are tautology or not.(using truth table)

(P→(┐P)) → (┐P) ii) (P→(Q→R)) →((P→Q) →(P→R))

Using Predicate, Quantifier and rule of inference determine the given argument is valid or not “All student in the class understand logic. Xavier is a student in this class. Therefore, Xavier understand logic.”

Each student in a class of 40 plays at least one indoor game chess, carrom and scrabble. 18 play chess, 20 play scrabble and 27 play carrom. 7 play chess and scrabble, 12 play scrabble and carrom and 4 play chess, carrom and scrabble. Find the number of students who play (i) chess and carrom. (ii) chess, carrom but not scrabble.

Let X = {1,2,3,4,5} R={<x,y> | x>y}. Draw a graph of Rand also give its matrix. Check whether the given relation an equivalence relation?

2 n(n1) 132333.........n3     2 

For an integer x prove that the following statements are equivalent: p: x is divisible by 10. q: x is divisible by 2 and 5. r: x is an even number and x is divisible by 5.

 1 2 3 If A   and 4 2 51 B   24  then find AB, BA. Show that AB 07 BA

Explain basic properties of integer with examples.

Let X = {1,2,3,4,5} and R,S,Tbe the relation as follows: R={(x,y)/x+y=5} S={(1,2),(3,4),(2,2)}T = {(4,2),(2,5),(3,1),(1,3)} (i) Write properties of R . (ii) Write matrix of R .(iii) Find SoT, Ro Sand So R .

Name different techniques of proof. Explain “the method of proof by contradiction”, giving suitable example.

Define Composition of a function. Let X={1,2,3} and f,g,hand s be functions from Xto X given by f = {<1,2>,<2,3>, <3,1>} g = {<1,2>,<2,1>, <3,3>} h = {<1,1>,<2,2>, <3,1>} s = {<1,1>, <2,2>, <3,3>} Find fog ,gof, fohog, sog, gos, sos.

Define node base of a diagraph. Find all node base of the digraph given below

Define: Isomorphic Graph. Verify the following graphs are isomorphic or not.(Justify)

Define Binary tree. Convert the given tree into the Binary tree (v0(v1(v2)(v3)(v4))(v5(v6)(v7)(v8)(v9))(v10(v11)(v12))).

Define adjacency matrix of a graph and obtain the adjacency matrix (A) for the following graph. Find AT. Also draw graph of AT and find Path matrix P.