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### High School Mathematics8.3 Types of Relations - Reflexive, Symmetric, Anti-Symmetric, Transitive, and Equivalance

 1. Reflexive Relation: R is a relation in A and for every a Î A, (a,a) Î R then R is said to be a reflexive relation. Example: Every real number is equal to itself. Therefor "is equal to " is a reflexive relation in the set of real numbers. 2. Symmetric Relation: R is a relation in A and (a,b) Î R implies (b,c) Î R then R is said to be a symmetric relation. Example: In the set of all real numbers "is equal to" relation is symmetric. 3. Anti-Symmetric Relation: R is a relation in A. If (a,b) Î R and (b,a) Î R implies a = b, then R is said to be an anti-symmetric relation. Example: In set of all natural numbers the relation R defined by "x divides y if and only if (x,y) Î R" is anti-symmetric. For x|y and y|x then x = y. 4. Transitive Relation: R is a relation in A if (a,b) Î R and (b,c) Î R implies (a,c) Î R is called a transitive relation. Example: In the set of all real numbers the relation "is equal to" is a transitive relation. For a = b, b = c implies a = c. 5. Equivalence Relation: A relation R in a set A is said to be an equivalence relation if it is reflexive, symmetric and transitive. Example: In the set of all real numbers the relation "is equal to" is an equivalence relation for a Î R, a = a, b = a implies b = a and a = b, b = c implies a = c. Directions: Choose the correct answer. Also write at least 10 examples of your own.
 Q 1: Every reflexive relation in a set A is an _____.Transitive relationAnti-symmetric relationAn Equivalence relationnot an Anti-symmetric relation Q 2: R is a relation in A. If (a,b) Î R and (b,c) Î R implies (a,c) Î R, then R is called a _______.Transitive relationAnti-symmetric relationSymmetric relationReflexive relation Q 3: If R is a relation in A and for every a Î A, (a,a) Î R then R is said to be a _______.Reflexive relationTransitive relationSymmetric relationAnti-Symmetric relation Q 4: R is a relation in A and (a,b) Î R implies (b,a) Î R then R is said to be a _______.Anti-symmetric relationReflexive relationSymmetric relationTransitive relation Question 5: This question is available to subscribers only! Question 6: This question is available to subscribers only!

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