Relation of One Set to Another

 ♨ Relation R is a subset of a cartesian product of A ⊗ B i.e.  R⊄ AxB . R' which is a subset of BxA is called an inverse relation denoted by R-1 ‡ Example: Set A={1,2,3} and Set B ={1,3,5} Then K=AxB ={(1,1),(1,3),(1,5),(2,1),(2,3),(2,5),(3,1),(3,3),(3,5)} ↹ Set M ={(1,1),(1,3),(3,1),(3,3)} is subset of K and therefore a relation of A →B ➰ If a Relation is defined between two sets, it is called a Binary Relation. ➲ A relation  R is said to be Reflexive if   aRa holds true for ∀a ⊄ M . M is a reflexive relation because of (1,1) & (3,3) ➲ A relation  R  is said to be symmetric  if aRb ⇒ bRa ∀a,b ⊄ M. Here, (1,3) & (3,1) both belong to M. M is a symmetric relation because of (1,1) & (3,1) ♦ A relation is said to be transitive if aRb, bRc ⇒ aRc ∀a,b,c ⊄ M . Here M is not a transitive relation. ♦ A relation which is reflexive, symmetric as well as transitive is called an Equivalent Relation. ♦ A relation which is reflexive, anti-symmetric as well as transitive is called an Partial Ordered Relation. ♦ Relation Matrix: In a binary relation, matrix elements mij = 1 iff aiRaj holds good, otherwise  mij = 0. If R ⊄ AxA, then relation matrix is a square matrix of the same order as that of A. Otherwise, it is a rectangular matrix of m rows and n columns where m is the order of A and n is the order of B. ♦ Example: Suppose R consists of ordered set of integers (x,y) such that (x,y) ∈ R & x2 + y2 =25 . Then R={(-5,0),(0,-5),(0.5),(5,0),(3,4),(4,3),(-3,4),(4,-3),(-4,3),(3,-4),(-3,-4),(-4,-3)} and A= {-5,-4,-3,0,3,4,5} . Order of A is 7 and therefore no. of elements of relation matrix will be 49. ♦ The diagonal elements m11,m22,m33,m4,m55,m66,m77 are equal to zero. m14=m41=1, m23=m32=1, m25=m52=1,m36=m63=1, m47=m74=1,m56=m65=1. Rest are all zero. Here R=R-1 ♦ It is a symmetric matrix if both set A and B have the same cardinality, otherwise it is a rectangular matrix. Working- Put 2 sets of same cardinality. write the relations. click 1. Then click the image of any relation and then click-3. repeat this for all images. The images u have finished will be yellow.
 RELATION MATRIX cardinality A: cardinality B: cardinality A ⊗ B ↦ set A set B matrice mij -- Relation Set -- -1 -- -2 -- -3 -- -4 -- -5 -- -6 -- -7 Elements -- -8 -- -9 -- -10 -- -11 -- -12 -- -13 -- -14 -- -15 -- -16 -- -17 -- -18 -- -19 -- -20 -- -21 -- -22 -- -23 -- -24 -- -25 -- -26 -- -27 -- -28