Discrete Mathematics Answers

Let 𝑄(𝑥, 𝑦) denote "𝑥 + 𝑦 = 𝑦“. What are the truth values of the quantifications ∃𝑦∀𝑥𝑄(𝑥, 𝑦) and ∀𝑥∃𝑦𝑄(𝑥, 𝑦) where the domain for all variables consists of all real numbers?

Show that A = B where A = {x│x² - 4x +4 =1}, B = {1,3}

List all strings over X = {0,1}  of length 2 or less.

Using a truth table, prove or disprove the following:

~[(~p∧q)↔ r]≡ ~(p∨~q)↔ ~r

B.Write the following sets using rule method:

6-7. Write A= {10, 20, 30, 40, 50} using rule method.

8-9. Write C={𝑥|𝑥 𝑖𝑠 𝑎 province in the region of CALABARZON} using roster method

A. Rewrite the following statements using proper set notation.

1. x is a member of A

2. d is not an element of E

3. M and N are equal sets

4. B is the set of all x such that x squared is equal to 9

5. J is a subset of K

C. Given A= {d}, B= {c, d}, C= {a, b, c}, D = {a, b}, E= {a, b, d}. Determine whether the following statements are True or False. 10. 𝐴 ⊂ 𝐵

11. 𝐶~𝐷

12. 𝐷 ⊈ 𝐸

13. n(C) =3

14. C and D are joint sets

15. E has 16 subsets D. Consider the following sets:

1. Show that in any set of six classes, each meeting regularly once a week on a particular day of the week, there must be two that meet on the same day, assuming that no classes are held on weekends.

2. Show that if there are 30 students in a class, then at least two have last names that begin with the same letter.

{(1,2), (3,5)} is a partition of {1,2,3,4,5}?

Out of 300 students taking discrete mathematics 60 take coffee , 27 take coco, 36 take tea, 17 take tea only, 47 take chocolate only, 7 take chocolate and coco, 3 take chocolate tea and coco, 20 take coco only, 2 take tea, chocolate and coco, 30 take coffee only, 9 take tea and chocolate whereas 12 take 12 tea and coffe

Express this information on venn diagram

Find how many take beverages

Find how many take tea and coco