Without proving anything, determine if the following statements are true or false. For any false statements, give an counterexample. (a) A finite, nonempty set of real numbers always contains its supremum. (b) If a < L for every element a in the set A, then sup A < L. (c) If A and B are sets with the property that a < b for every a € A and b e B, then sup A < inf B. (d) If sup A = s and sup B = t, then sup(A+B) = s+t. Here and elsewhere, the set A+B is defined as A+B = {a+b:a € A, and b e B}. (e) If sup A < sup B then there is an element of B that is an upper bound for A.

Determine, including a proof, the infimum of the set B = {x ∈ R ?? |x−1| < 4}. You may want to use the fact that “|z| < w if and only if −w < z < w” for any expressions z and w.

Let the universal set be R, the set of all real numbers, and let A = {x € R | −3 ≤ x ≤ 0}, B = {x € R | −1 < x < 2}, and C= {x € R | 6 < x ≤ 8}. Find each of the following: a. AUB b. An B c. Ac d. AUC e. Anc f. Bc g. An Be

Prove whether this statement is true or false. For all sets A, B, and C: (see image)

(9) Is A x BCR? Why or why not? B. Let N be the set of natural numbers, A = {1,...), B = {:1≤ ≤ 10, where z €N}. (10) State explicitly the elements of the set B (write set B in set-roster notation). (11) Write set A in set-builder notation (follow the format A = {re X: P(x)}). For items (12)-(14), Let the mapping F: N→A be defined as F(x) = ¹. (12) Evaluate F (10)

1MATH 11

Let U = (−∞, ∞) be the universal set and M = (−6, 5] and W = (−3, 9]. In interval notation, determine the following. 1. M ∪ W2. M ∩ W 3. M − W4. W − M

Help me please

let sets a and b be defined as follows. A is the set of integers than or equal -5 and less than or equal to -1