1. **Problem:** Check whether the relation $R = \{(1,2),(2,2),(1,1),(1,3),(3,2),(3,3),(4,4)\}$ on the set $\{1,2,3,4\}$ is reflexive, symmetric, or transitive. 2. **Reflexive:** A relation $R$ on set $A$ is reflexive if $\forall a \in A, (a,a) \in R$. 3. Check if $(1,1), (2,2), (3,3), (4,4) \in R$. 4. Given $R$ contains $(1,1), (2,2), (3,3), (4,4)$, so $R$ is reflexive. 5. **Symmetric:** $R$ is symmetric if whenever $(a,b) \in R$, then $(b,a) \in R$. 6. Check pairs: $(1,2) \in R$ but $(2,1) \notin R$, so $R$ is not symmetric. 7. **Transitive:** $R$ is transitive if whenever $(a,b) \in R$ and $(b,c) \in R$, then $(a,c) \in R$. 8. Check pairs: $(1,2)$ and $(2,2)$ imply $(1,2)$ in $R$ (true), $(1,3)$ and $(3,2)$ imply $(1,2)$ in $R$ (true), etc. 9. All such pairs satisfy transitivity, so $R$ is transitive. **Answer:** $R$ is reflexive and transitive but not symmetric. 2. **Problem:** Show that $R = \{(a,b) : a \leq b\}$ on $\mathbb{R}$ is reflexive and transitive but not symmetric. 3. Reflexive: $a \leq a$ is true for all $a$, so $R$ is reflexive. 4. Transitive: If $a \leq b$ and $b \leq c$, then $a \leq c$, so $R$ is transitive. 5. Symmetric: If $a \leq b$, generally $b \leq a$ is false unless $a=b$, so $R$ is not symmetric. 3. **Problem:** Determine if $R = \{(x,y) : y = x + 5, x < 4\}$ on $\mathbb{N}$ is reflexive, symmetric, transitive. 4. Reflexive: For $x$, $(x,x)$ requires $x = x + 5$, impossible, so not reflexive. 5. Symmetric: If $(x,y) \in R$, then $y = x + 5$, but $(y,x)$ requires $x = y + 5$, generally false, so not symmetric. 6. Transitive: $(x,y)$ and $(y,z)$ imply $y = x + 5$ and $z = y + 5 = x + 10$, but $(x,z)$ requires $z = x + 5$, false, so not transitive. 4. **Problem:** Show $R = \{(P_1,P_2) : P_1, P_2$ have same number of sides$\}$ on polygons $A$ is equivalence relation. 5. Reflexive: Every polygon has same number of sides as itself. 6. Symmetric: If $P_1$ has same sides as $P_2$, then $P_2$ has same sides as $P_1$. 7. Transitive: If $P_1$ same sides as $P_2$ and $P_2$ same sides as $P_3$, then $P_1$ same sides as $P_3$. 8. So $R$ is equivalence relation. 9. The right angle triangle $T$ has 3 sides, so all polygons with 3 sides (triangles) are related to $T$. 5. **Problem:** Number of equivalence relations on $\{1,2,3\}$ containing $(1,2)$ and $(2,1)$ is 2. 6. Since $(1,2)$ and $(2,1)$ are in $R$, $1$ and $2$ are in same equivalence class. 7. Equivalence relations partition the set. Possible partitions: - $\{\{1,2\}, \{3\}\}$ - $\{\{1,2,3\}\}$ 8. So, two equivalence relations contain $(1,2)$ and $(2,1)$. 6. **Problem:** Prove $f: \mathbb{R} \to \mathbb{R}$ defined by $4x^2 + 12x + 15 = 0$ is one-one. 7. This is a quadratic equation, but question likely means $f(x) = 4x^2 + 12x + 15$. 8. To check one-one, check if $f(x_1) = f(x_2)$ implies $x_1 = x_2$. 9. $f(x_1) = f(x_2) \Rightarrow 4x_1^2 + 12x_1 + 15 = 4x_2^2 + 12x_2 + 15$. 10. Simplify: $4(x_1^2 - x_2^2) + 12(x_1 - x_2) = 0$. 11. Factor: $4(x_1 - x_2)(x_1 + x_2) + 12(x_1 - x_2) = 0$. 12. Factor out $(x_1 - x_2)$: $(x_1 - x_2)(4(x_1 + x_2) + 12) = 0$. 13. So, either $x_1 = x_2$ or $4(x_1 + x_2) + 12 = 0$. 14. The second condition can hold for distinct $x_1, x_2$, so $f$ is not one-one. 15. Hence, $f$ is not one-one. 7. **Problem:** If $f(x) = (3 - x^3)^{1/3}$, find $(f \circ f)(x)$. 8. Compute $f(f(x)) = f\big((3 - x^3)^{1/3}\big) = \big(3 - (3 - x^3)\big)^{1/3} = (x^3)^{1/3} = x$. 8. **Problem:** Is $g = \{(1,1),(2,3),(3,5),(4,7)\}$ a function? If $g(x) = \alpha x + \beta$, find $\alpha, \beta$. 9. $g$ is a function since each input has exactly one output. 10. Use points $(1,1)$ and $(2,3)$: $1 = \alpha \cdot 1 + \beta$ and $3 = \alpha \cdot 2 + \beta$. 11. Subtract: $3 - 1 = 2 = 2\alpha - \alpha = \alpha$. 12. So, $\alpha = 2$. 13. From $1 = 2 + \beta$, $\beta = -1$. 9. **Problem:** Find range of $f(x) = \frac{1}{2} - \cos x$. 10. Since $\cos x \in [-1,1]$, 11. $f(x) = \frac{1}{2} - \cos x$ ranges from $\frac{1}{2} - 1 = -\frac{1}{2}$ to $\frac{1}{2} - (-1) = \frac{3}{2}$. 12. So, range is $[-\frac{1}{2}, \frac{3}{2}]$. 10. **Problem:** Examples of mappings: (i) One-one but not onto: $f: \mathbb{N} \to \mathbb{N}$ defined by $f(x) = x + 1$. (ii) Not one-one but onto: $f: \{1,2\} \to \{a\}$ defined by $f(1) = a$, $f(2) = a$. (iii) Neither one-one nor onto: $f: \{1,2\} \to \{a,b,c\}$ defined by $f(1) = a$, $f(2) = a$ (not onto $b,c$ and not one-one).