Task 1. Sets and Counterexamples 1. For each of the following statements about sets A, B and C, give a counterexample to show that the statement is false. (a) If A E B and B C C, then ACC. (b) If A E B and B E C, then A E C. Task 2. Set Operations 1. Consider the symmetric difference of the sets A and B, which is defined as: AAB := (AUB) - (ANB) to answer the following questions. (a) Using a Venn Diagram, represent AAB. (b) Is the operation A commutative? Why or why not? (c) Give a sufficient and necessary condition for AAB = AUB. 2. For every iЄ N, A₁ = (-1, ½), where A; denotes the open interval from - to. =: (a) Determine U1 A₁ = A₁ UA₂ U A3 and 1 A₁ = A₁ NA₂ NA3. Provide an explanation for your answer. (b) Determine U₁₁₁ and Ai. Provide an explanation for your answer. Note: ∞ represents infinity. Task 3. Proofs 1. Prove or disprove (AB) = AUB. 2. Prove the following statements directly from the definitions of the terms. Do not use any other facts previously proved in class or in the textbooks or in the exercises. (a) Vn Є Z, n² + n is even. (b) Va, b, c Z, if a|b and a/c, then a|(56+ 3c).

For what values of c is c(1,2,3)=1?

Does the relation is in love with have a reflexive property consider one person? a symmetric property consider two people? a transitive property consider three people?

7. For the given subsets and of Z, let and determine whether is onto and whether it is one-to-one. Justify all negative answers. a. b. c. d.

Are these points collinear? a A-2,5, B0,2, and C4,-4 b D-1,-1, E2,-2, and F5,-5

Label each of the following statement as either true or false. only if

Q14C. Define the sets A, B, C, and D as follows: A = {-3, 1, 6, 7, 10} B = {-10, -5, 1, 4, 6} C = {x E Z: x is odd} D = {x E Z: x is positive} Find B n C. *Express the set using roster notation and with values ordered smallest to largest, such as {-10, -3, 1, 6} (braces and commas are required; a space after each comma is also required).

Use any method you choose to show that An (BU C) = A U B n C, when A, B, and C are sets.

2. Which of the following could be a linear combination of the set {(a, b), (c, d)}? (Circle ALL that apply) a. (3a, 3b) (2a + 2c, b + d) d. (2a-c, 2b-d) b. (2a, b + 2d) C.

Q14C. Define the sets A, B, C, and D as follows: A = {-3, 1, 6, 7, 10} B = {-10, -5, 1, 4, 6} C = {x ∈ Z: x is odd} D = {x ∈ Z: x is positive} Find B ∩ C.  *Express the set using roster notation and with values ordered smallest to largest, such as {-10, -3, 1, 6} (braces and commas are required; a space after each comma is also required).