need help answering 6-10 for my discrete mathematics class. 6. Translate each of

Question

need help answering 6-10 for my discrete mathematics class.

6. Translate each of these quantifications into English and determine its truth value a) 3x E R(x3 -1) b) 3x E Z(x 1 x) c) Vx E Z(x 1 E Z) 7. a) Find union of (1, 5, 6 and 12, 5, 8) b) Find intersection of {1, 3, 4, 5 and 12, 4, 5, 6h c) The binary string for sets 12, 4, 6, 8, 10) and (1, 2, 3, 4, 5) are 0101010101 and 1111100000 respectively. Find union and intersection of the sets. d) Let A, B and C be sets. Showthat (B-A) U(C-A)E (BUC) -A

Explanation / Answer

8a.  A C = B C

The answer is No.

Lets assume
A = {1, 2, 3}
B = {1, 2, 3, 4}
C = {1, 2, 3, 4, 5}
A C = B C = {1, 2, 3, 4, 5}, but A is not equal to B.

_____________________________________________________

8b. A C = B C  

The answer is no. Here's a counterexample:

Lets consider,

A = {1, 2, 3}
B = {1, 2, 3, 4}
C = {1, 2}

A C = B C = {1, 2}, but A is not equal to B.

______________________________________________

8c.  A C = B C and A C = B C

The answer is Yes.

To prove that A = B, we can prove that A is a subset of B, and prove that B is a subset of A.

Assume temporarily that A is not a subset of B.

Then there exists an element p that is in A set but not in B set.

If p is not in C set, then p is in A C but p is not in B C, contradicting A C = B C.

If p is in C, then p is in A C but p is not in B C, contradicting A C = B C.

So whether p is in C or not in C, we obtain a contradiction.

Therefore, our assumption that A is not a subset of B is a false statement; hence we conclude that A is a subset of B.

By a similar argument switching sets A and B, we conclude that B is a subset of A.

Since A and B are subsets of each other, we finally conclude that A = B.

Related Questions

Navigate

• Browse (All)
• Browse N
• Subjects

---

---

Integrity-first tutoring: explanations and feedback only — we do not complete graded work. Learn more.