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sunainagupta
Consider a set X = [2, 3, 4) and the Relation defined on X by. R = {(2, 2) (2, 3) (3, 3) (3, 4) (2, 4) (4, 4)}. Find whether R is : i) Reflexive ii) Symmetric iii) Transitive Also justify your answer.
Reflexive means: \[ \forall x \quad xRx \]
It's reflexive since it contains \((2,2)\), \((3,3)\), and \((4,4)\).
Symmetric means: \[ \forall x\quad xRy\iff yRx \]
It's not symmetric because it contains \((2,3)\) but not \((3,2)\).
@sunainagupta Are you following?
Transitive means: \[ \forall x\quad xRy\wedge yRz\implies xRz \]
Let me rehash, since my notation was strange before. Reflexive \[ \forall x \quad (x,x)\in \mathcal{R} \] Symmetric \[ \forall x,y \quad (x,y)\in \mathcal{R} \iff (y,x)\in \mathcal{R} \] Transitive \[ \forall x,y,z \quad (x,y)\in \mathcal{R} \wedge (y,z)\in \mathcal{R} \implies (x,z)\in \mathcal{R} \]
A survey among the students of college. 60 Study Hindi, 40 study Spanish, and 45 study Japanese, Further 20 study Hindi and Spanish, 25 study Hindi and Japanese, 15 study Spanish and Japanese and 8 study all the languages. Find the followings: i) How many students are studying at least one language? ii) How many students are studying only Hindi ? iii) How many students are studying only Japanese ?