14h - Discrete Mathematics Equivalences 14-2 Previous...

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Introduction Equivalences Discrete Mathematics Andrei Bulatov Discrete Mathematics – Equivalences 14-2 Previous Lecture Cartesian products of two and more sets Cardinality and other properties of Cartesian products Binary, ternary and k-ary relations Describing binary relations Discrete Mathematics – Equivalences 14-3 Properties of Binary Relations – Reflexivity From now on we consider only binary relations from a set A to the same set A. That is such relations are subsets of A × A. A binary relation R A × A is said to be reflexive if (a,a) R for all a A. (a,b) R Z × Z if and only if a b This relation is reflexive, because a a for all a Z Matrix: 1 * * * * 1 * * * * 1 * * * * 1 1’s on the diagonal Graph: Loops at every vertex Discrete Mathematics – Equivalences 14-4 Properties of Binary Relations – Symmetricity A binary relation R A × A is said to be symmetric if, for any a,b A, if (a,b) R then (b,a) R. The relation Brotherhood (`x is a brother of y’) on the set of men is symmetric, because if a is a brother of b then b is a brother of a Matrix: 1 1 1 1 Matrix is symmetric w.r.t. the diagonal Graph: Graph is symmetric Discrete Mathematics – Equivalences 14-5 Properties of Binary Relations – Transitivity A binary relation R A × A
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