Permutations & Combinations
Relations on set A={1,...,26} — asymmetric, reflexive-not-symmetric, etc.
MJAT_TS8_P1
Grade 12
Question:
Let $A=\{1,2,\ldots,26\}$. Match each type of relation on $A$ with the count formula and find $p+q$, $p+q+r$, etc.
**List-I:** P) Asymmetric relations=$p^q$ ($p$ prime), find $p+q$; Q) Reflexive but not symmetric $=p^q(p^q-p^r)$, find $p+q+r$; R) Symmetric but not reflexive (non-empty) $=p^q(p^r-1)-1$, find $p+q+r$; S) Neither symmetric nor reflexive (non-empty) $=(p^s-1)p^q(p^q-1)$, find $p+q+s$
**List-II:** 1)328; 2)329; 3)330; 4)353; 5)356
A) P→1; Q→1; R→4; S→4
B) P→1; Q→2; R→3; S→4
C) P→3; Q→2; R→5; S→5
D) P→3; Q→1; R→4; S→5
Step-by-Step Solution
Key Concept: P: Asymmetric = $3^{325}$ (each non-diagonal pair can be: neither, $(a,b)$, $(b,a)$ = 3 choices; $|A|^2-|A|=650$ off-diagonal, in 325 unordered pairs). $p=3$, $q=325$, $p+q=328\to$ list 1. Q: Reflexive not symmetric: from $2^{650}-2^{325}=2^{325}(2^{325}-1)$: $p+q+r=2+325+325=652$? Hmm.
Answer: **A**.
Correct Answer: A