1. **Problem statement:** Classify the propositions as true (V) or false (F) and justify the false ones. 2. **Given data:** Correspondence table: $$\begin{array}{c|cccccc} A & -1 & 0 & 1 & \frac{1}{2} & \sqrt{2} & 2^{-1} \\ \hline B & -1 & 0 & 3 & 4 & 5 & 7 \end{array}$$ 3. **Step 1: Analyze proposition a)** "The correspondence defined by the table is a function from $A$ to $B$." - A function assigns exactly one output in $B$ for each input in $A$. - Each element in $A$ has one unique image in $B$. - From the table, each $A$ value corresponds to exactly one $B$ value. - Therefore, proposition a) is **True (V)**. 4. **Step 2: Analyze proposition b)** "If $A$ and $B$ are numeric sets such that $A \subseteq B$, then..." (incomplete statement, so cannot classify). - Since the statement is incomplete, we cannot determine its truth value. 5. **Step 3: Analyze proposition c)** "The values of $x$ for which the algebraic fraction is zero are 0." - Without the explicit fraction, assume the fraction is $\frac{P(x)}{Q(x)}$. - The fraction is zero when $P(x) = 0$ and $Q(x) \neq 0$. - Since no fraction is given, this proposition cannot be verified. - Therefore, proposition c) is **False (F)** due to lack of information. 6. **Step 4: Analyze proposition d)** "The solution set of the inequality is $\ldots$" (incomplete). - Without the inequality, cannot classify. 7. **Step 5: Calculate the slope of the line passing through points $A(4,0)$ and $B(0,3)$** Formula for slope: $$m = \frac{y_2 - y_1}{x_2 - x_1}$$ Substitute: $$m = \frac{3 - 0}{0 - 4} = \frac{3}{-4} = -\frac{3}{4}$$ 8. **Step 6: Calculate the length of segment $AB$** Distance formula: $$d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}$$ Substitute: $$d = \sqrt{(0 - 4)^2 + (3 - 0)^2} = \sqrt{(-4)^2 + 3^2} = \sqrt{16 + 9} = \sqrt{25} = 5$$ 9. **Step 7: Triangle formed by points $A$, $B$, and origin $O(0,0)$** - Right triangle with legs along axes. - Angles at $O$ is $90^\circ$. - Other angles can be found using trigonometry: $$\theta = \arctan\left(\frac{3}{4}\right)$$ **Final answers:** - a) V - b) Cannot classify (incomplete) - c) F (no fraction given) - d) Cannot classify (incomplete) - Slope of line $AB$ is $-\frac{3}{4}$ - Length of segment $AB$ is 5 - Triangle angles: right angle at origin, others from $\arctan(3/4)$