The number of relations, defined on the set \a, b, c, d\ , which are both reflexive and symmetric, is equal to:

Solution & Explanation

### Related Formula For a set with n elements, the number of relations that are both reflexive and symmetric is given by: 2^fracn(n-1)2 ### Core Logic A reflexive relation must contain all diagonal pairs (x,x). There is only 1 way to assign these elements (they MUST be present). A symmetric relation requires that if (x,y) is present, (y,x) must also be present. Thus, we only have the freedom to choose whether to include the unordered pairs \x, y\ where x neq y. ### Step 1: Calculate the available independent pairs Number of distinct elements n = 4. Total number of pairs in the cartesian product is n^2 = 16. Number of diagonal pairs (reflexive necessity) = n = 4. Remaining non-diagonal pairs = 16 - 4 = 12. Since symmetry pairs them up (a,b) leftrightarrow (b,a), there are exactly frac122 = 6 independent choices. ### Step 2: Calculate total relations Each of the 6 independent pairs can either be included or excluded (2 choices). Total relations = 1^4 times 2^6 = 64. ### Pattern Recognition Memorize the combinatorics of binary relations for n elements: Total = 2^n^2, Reflexive = 2^n(n-1), Symmetric = 2^n(n+1)/2, Reflexive & Symmetric = 2^n(n-1)/2. ### Evaluation Rubric / Model Answer null ### Chapter Mix Class 12 Maths: Sets and Relations

Reference Study Guides

Sets And Relations
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