The following are two proofs that for all sets A and B , A − B ⊆ A . The first is less formal, and the second is more formal. Fill on the blanks. Proof: Suppose A and B are any sets. To show that A − B ⊆ A , we must show that every element in (1) is in (2) But any element in A - B is in (2) But any element in A - B is in (3) and not in (4) (by definition of A - B ). In particular, such an element is in A . Proof: Suppose A and B are any sets and x ∈ A − B . [We must shoe that (1) J By definition of A - B ). In particular, such an element is in A.

Discrete Mathematics With Applicat...

5th Edition
EPP + 1 other
Publisher: Cengage Learning,
ISBN: 9781337694193

Chapter
Section

Discrete Mathematics With Applicat...

5th Edition
EPP + 1 other
Publisher: Cengage Learning,
ISBN: 9781337694193
Chapter 6.2, Problem 2ES
Textbook Problem
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The following are two proofs that for all sets A and B, A − B ⊆ A . The first is less formal, and the second is more formal. Fill on the blanks. Proof: Suppose A and B are any sets. To show that A − B ⊆ A , we must show that every element in (1) is in (2) But any element in A-B is in (2) But any element in A-B is in (3) and not in (4) (by definition of A-B). In particular, such an element is in A.Proof: Suppose A and B are any sets and x ∈ A − B . [We must shoe that (1) J By definition of A-B). In particular, such an element is in A.

To determine

(a)

Proof: Suppose A and B are any sets. To show that ABA, we must show that every element in __(1)__ is in __(2)__. But any element in AB is in ___(3)___ and not in ___(4)___(by definition of AB ). In particular, such an element is in A.

Explanation of Solution

Given information:

A and B are any sets.

Concept used:

Any element AB is in A and not in B

Calculation:

Suppose  A and B are two set.

To show that ABA, need to show that if xAB then xA.

That is, every element in AB is in A.

From the definition of subtraction of sets xABxA and xB.

So any element in AB is in A and not in B.

By this, conclude that xA

To determine

(b)

Suppose  A and B are any sets and xAB. [we must show that (1) ]. By definition of set difference, x(2) and x(3). In particular, x(4) [which is what was to be shown].

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