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1.Relation and Function
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मान लें $A, 10$ अवयवों वाला एक समुच्चय है. $A$ से $A$ में अतिरिक्त संबंधों की संख्या जो स्वतुल्य $(reflexive)$ हैं परन्तु सममित $(symmetric)$ नहीं है, कितनी होगी?
A
$2^{89}-1$
B
$2^{89}-2^{45}$
C
$2^{45}-1$
D
$2^{90}-2^{45}$
(KVPY-2020)
Solution
(d)
Since, $A \times A$ contains $100$ ordered pairs $(a, b)$ out of which $10$ ordered pairs are such that $a=b$.
For a reflexive relation $(a, a)$ must be present and others have a choice of to be present or not.
So, number of reflexive relations $=2^{90}$.
For a symmetric relation if $(a, b)$ is present then $(b, a)$ is also present
(where $a \neq b$ ). There are $45$ such pairs of ordered pairs.
So, number of reflexive relations which are also symmetric $=2^{45}$
$\therefore$ Required number of relations $=2^{90}-2^{45}$.
Standard 12
Mathematics
