In set theory, what is the duality principle, and how does it apply to operations like union and intersection?

You will need to have some squares, you can cut out paper squares that are 1 inch by 1 inch, use cheez its, use starburst, saltine crackers, or any other object that is a square. Choose a certain number of squares that you want to work with - only restriction is that you must use at least 7 squares. Create a shape using the number of squares that you have chosen. This can be any shape that you are able to create with your set number of squares. Show how to determine the Area of the shape that you created - DO NOT use the Area formulas. Using the same shape, show how to determine the Perimeter of that shape you created - again DO NOT use the Perimeter formulas. Then take the squares that you are using and create a NEW shape. Show how to determine the Area of this new shape. Show how to determine the Perimeter of this new shape. Be sure to reference any differences or similarities that your students might notice about the two shapes.

A. Consider the axiomatic system as described. Axiom1: There are exactly three pins. Axiom2: Every pins is on atleast two lines. Axiom 3: Each line passes through at most two pins. 1. Identify the undefined terms and the undefined relations in the axiomatic system. For Items 2 and 3, explain why the given propositions cannot be considered as additional axioms for the system. Note that an axiomatic system should be consistent and independent. 2. "There are exactly two lines." Why cannot this proposition be an additional axiom in the given system? 3. "There are at least three lines." Why cannot this proposition be an additional axiom in the given system? B. Consider a ABC Student Axiomatic System as described. undefined terms: ABC student, scholarly person, intelligent person, honorable person, excellent person Axiom 1: All ABC students are intelligent persons. Axiom2: Some ABC students are honorable persons. Axiom 3: All intelligent persons are scholarly persons. Axiom 4: Excellent…

Consider a popular ramen stall. Before the shop opens, there are 20people waiting in front of the restaurant.    Given that the shop only serves 10 bowls per day. If the maximum number ofeach menu the customers can order is 7 which means if the first 7 people orderShoyu Ramen, the rest must order Seafood Ramen. How many combinationof ramen orders(how each customer order and who got kicked out) are there?Ex: People can still order 5 Shoyu and 5 Seafood if they wish. But, no, the10 people who gets in can’t order 8 Shoyu and 2 Seafood.