1. **Problem:** Determine if the statement $P \lor (P \leftrightarrow Q) \lor Q$ is a tautology or contradiction using a truth table. 2. **Problem:** Calculate the cross product of two vectors with magnitudes $|a|=2\sqrt{3}$ and $|b|=3$, and angle between them $53^\circ$. 3. **Problem:** Find $\mathbf{a} + 2\mathbf{b} - \mathbf{c}$ where $\mathbf{a} = 2\mathbf{i} + 5\mathbf{j}$, $\mathbf{b} = 12\mathbf{i} - 10\mathbf{j}$, $\mathbf{c} = -3\mathbf{i} + 9\mathbf{j}$ using column matrix notation. 4. **Problem:** Given $2A - 3B = 4C$ with matrices $A$, $B$, and $C$ having unknowns $a,b,c$, find these variables. 5. **Problem:** Given vectors $\mathbf{a}$ and $\mathbf{b}$ with magnitudes $|\mathbf{a}|$ unknown, $|\mathbf{b}|=6$, and dot product $\mathbf{a} \cdot \mathbf{b} = $ unknown, find the angle between them or state if it does not exist. 6. **Problem:** Given $n(U)=169$, $n(A)=81$, $n(B)=96$, and $n(A \cup B)=137$, find $n(A \cap B)$ and draw a Venn diagram. --- ### Step 1: Truth Table for $P \lor (P \leftrightarrow Q) \lor Q$ - The biconditional $P \leftrightarrow Q$ is true when $P$ and $Q$ have the same truth value. - Construct truth table for $P$, $Q$, $P \leftrightarrow Q$, then evaluate $P \lor (P \leftrightarrow Q) \lor Q$. | $P$ | $Q$ | $P \leftrightarrow Q$ | $P \lor (P \leftrightarrow Q) \lor Q$ | |-----|-----|-----------------------|-------------------------------------| | T | T | T | T | | T | F | F | T | | F | T | F | T | | F | F | T | T | Since the expression is true for all truth values, it is a **tautology**. --- ### Step 2: Cross Product Magnitude Formula for magnitude of cross product: $$|\mathbf{a} \times \mathbf{b}| = |\mathbf{a}| |\mathbf{b}| \sin \theta$$ Given: $$|\mathbf{a}| = 2\sqrt{3}, \quad |\mathbf{b}| = 3, \quad \theta = 53^\circ$$ Calculate: $$|\mathbf{a} \times \mathbf{b}| = 2\sqrt{3} \times 3 \times \sin 53^\circ = 6\sqrt{3} \times 0.7986 \approx 6 \times 1.732 \times 0.7986$$ $$\approx 6 \times 1.384 = 8.304$$ So, the magnitude of the cross product is approximately **8.3**. --- ### Step 3: Vector Addition and Subtraction in Column Matrix Form Vectors: $$\mathbf{a} = \begin{bmatrix} 2 \\ 5 \end{bmatrix}, \quad \mathbf{b} = \begin{bmatrix} 12 \\ -10 \end{bmatrix}, \quad \mathbf{c} = \begin{bmatrix} -3 \\ 9 \end{bmatrix}$$ Calculate: $$\mathbf{a} + 2\mathbf{b} - \mathbf{c} = \begin{bmatrix} 2 \\ 5 \end{bmatrix} + 2 \begin{bmatrix} 12 \\ -10 \end{bmatrix} - \begin{bmatrix} -3 \\ 9 \end{bmatrix}$$ $$= \begin{bmatrix} 2 \\ 5 \end{bmatrix} + \begin{bmatrix} 24 \\ -20 \end{bmatrix} - \begin{bmatrix} -3 \\ 9 \end{bmatrix} = \begin{bmatrix} 2 + 24 + 3 \\ 5 - 20 - 9 \end{bmatrix} = \begin{bmatrix} 29 \\ -24 \end{bmatrix}$$ Final result: $$\begin{bmatrix} 29 \\ -24 \end{bmatrix}$$ --- ### Step 4: Solve for $a,b,c$ in $2A - 3B = 4C$ Assuming matrices: $$A = \begin{bmatrix} a & b \\ c & d \end{bmatrix}, B = \begin{bmatrix} e & f \\ g & h \end{bmatrix}, C = \begin{bmatrix} i & j \\ k & l \end{bmatrix}$$ Given equation: $$2A - 3B = 4C$$ Equate each element: $$2a - 3e = 4i, \quad 2b - 3f = 4j, \quad 2c - 3g = 4k, \quad 2d - 3h = 4l$$ Solve for $a,b,c$ if $e,f,g,i,j,k$ are known or provide values if given. Since values are missing, this is the general form. --- ### Step 5: Find Angle Between Vectors Formula for dot product: $$\mathbf{a} \cdot \mathbf{b} = |\mathbf{a}| |\mathbf{b}| \cos \theta$$ Given: $$|\mathbf{b}| = 6, \quad \mathbf{a} \cdot \mathbf{b} = d$$ If $d$ and $|\mathbf{a}|$ are known, then: $$\cos \theta = \frac{\mathbf{a} \cdot \mathbf{b}}{|\mathbf{a}| |\mathbf{b}|}$$ If $|\cos \theta| \leq 1$, angle exists: $$\theta = \cos^{-1} \left( \frac{d}{|\mathbf{a}| \times 6} \right)$$ If $|\cos \theta| > 1$, no real angle exists. --- ### Step 6: Find $n(A \cap B)$ and Venn Diagram Formula: $$n(A \cup B) = n(A) + n(B) - n(A \cap B)$$ Given: $$n(U) = 169, n(A) = 81, n(B) = 96, n(A \cup B) = 137$$ Calculate: $$n(A \cap B) = n(A) + n(B) - n(A \cup B) = 81 + 96 - 137 = 177 - 137 = 40$$ So, $n(A \cap B) = 40$. The Venn diagram would show: - Universal set $U$ with 169 elements. - Circle $A$ with 81 elements. - Circle $B$ with 96 elements. - Intersection $A \cap B$ with 40 elements. --- **Final answers:** 1. Tautology 2. Cross product magnitude $\approx 8.3$ 3. $\begin{bmatrix} 29 \\ -24 \end{bmatrix}$ 4. General solution $2A - 3B = 4C$ element-wise 5. Angle $\theta = \cos^{-1} \left( \frac{\mathbf{a} \cdot \mathbf{b}}{|\mathbf{a}| |\mathbf{b}|} \right)$ if valid 6. $n(A \cap B) = 40$