Given: A set \( S = \{a, b, c, d\} \).
Step 1: Reflexive Relation
A relation is reflexive if for every element \( x \in S \), the pair \( (x, x) \) is included. So, the relation must include the following reflexive pairs:
\( (a, a), (b, b), (c, c), (d, d) \).
These are fixed and must always be in the relation.
Step 2: Symmetric Relation
A relation is symmetric if for every pair \( (x, y) \), if \( (x, y) \) is in the relation, then \( (y, x) \) must also be included.
For the non-reflexive pairs, we need to ensure symmetry:
For each of these 6 pairs, we have 2 choices:
Step 3: Total Number of Relations
Since there are 6 non-reflexive pairs, and for each pair, we have 2 choices, the total number of reflexive and symmetric relations is:
\( 2^6 = 64 \).
Final Answer: The number of reflexive and symmetric relations from \( S \to S \) is \( \boxed{64} \).
Let $R$ be a relation defined on the set $\{1,2,3,4\times\{1,2,3,4\}$ by \[ R=\{((a,b),(c,d)) : 2a+3b=3c+4d\} \] Then the number of elements in $R$ is
Let \(M = \{1, 2, 3, ....., 16\}\), if a relation R defined on set M such that R = \((x, y) : 4y = 5x – 3, x, y (\in) M\). How many elements should be added to R to make it symmetric.
In the given figure, the blocks $A$, $B$ and $C$ weigh $4\,\text{kg}$, $6\,\text{kg}$ and $8\,\text{kg}$ respectively. The coefficient of sliding friction between any two surfaces is $0.5$. The force $\vec{F}$ required to slide the block $C$ with constant speed is ___ N.
(Given: $g = 10\,\text{m s}^{-2}$) 
Two circular discs of radius \(10\) cm each are joined at their centres by a rod, as shown in the figure. The length of the rod is \(30\) cm and its mass is \(600\) g. The mass of each disc is also \(600\) g. If the applied torque between the two discs is \(43\times10^{-7}\) dyne·cm, then the angular acceleration of the system about the given axis \(AB\) is ________ rad s\(^{-2}\).
