Prove that for three sets \( \mathrm{A}, \mathrm{B} \) and C . \( (A \cap B)^{\prime}=A^{\prime} \cup B^{\prime} \) \( (A \cup B)^{\prime}=A^{\prime} \cap B^{\prime} \) Define Cartesian nroduct of two sets.

14. Which of the following statement is false? A. \( \{\emptyset\} \in\{0,\{\varnothing\}\} \) B. \( \emptyset \subseteq\} \) C. \( \{\emptyset\}=\{ \} \) D. \( \emptyset \subseteq\{0,\{\varnothing\}\} \)

Нзобразить корни 2 степени из \( \frac{1}{2}+i \cdot \frac{\sqrt{3}}{2} \) на комплексной плоскости.

You run an animal shelter facility and are asked to purchase food supply for the next month. You normally purchase two types of food, Purina and Natural Choice. Purina costs \( \$ 180 \) per bag. Each bag weighs 80 pounds and can feed as many as six animals per month. The Natural Choice bag costs \( \$ 110 \), weighs 60 pounds, and can feed four animals per month. There is a new type of food, Pedigree, which you must consider purchasing. Pedigree food costs \( \$ 110 \) per bag. Each bag of Pedigree weighs 180 pounds and can feed as many as 16 animals per month. You have been given a maximum budget of \( \$ 2,800 \) for this purchase, although you don't have to spend that much. The facility has room for no more than 2,000 pounds. You can only buy whole bags of each food. Create an adequate mode and solve it in Excel to maximize the number of animals to be fed during the month.

1. Two of the Eigen values of the matrix \( A=\left[\begin{array}{lll}2 & 2 & 1 \\ 1 & 3 & 1 \\ 1 & 2 & 2\end{array}\right] \) are equal to 1 each. Then solve the Eigen values of \( A^{-1} \).

points. 4. If \( u=3 i+6 j-7 k, v=4 i-5 j+6 k \) and \( w=-2 i+3 j-9 k \) then show that \( (u \times v) \bullet w=(v \times w) \bullet u=(w \times u) \bullet v \)