   # In 3 and 4, supply explanations of the steps in the given proofs. Theorem: For all sets A, B, and C , if A ⊆ B , B ⊆ C then A ⊆ C . Proof: Statement Explanation Suppose A, B, and C are any sets such that A ⊆ B and B ⊆ C . starting point We must show that A ⊆ C . conclusion to be shown Let x be any element in A. start of an element proof Then x is in B . ( a ) . It follows that x is in C. ( b ) . Thus every element in A is in C . since x could be any element of A Therefore, A ⊆ C [as was to be shown]. ( c ) . ### Discrete Mathematics With Applicat...

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

#### Solutions

Chapter
Section ### Discrete Mathematics With Applicat...

5th Edition
EPP + 1 other
Publisher: Cengage Learning,
ISBN: 9781337694193
Chapter 6.2, Problem 3ES
Textbook Problem
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## In 3 and 4, supply explanations of the steps in the given proofs. Theorem: For all sets A, B, and C, if A ⊆ B , B ⊆ C then A ⊆ C . Proof: Statement Explanation Suppose A, B, and C are any sets such that A ⊆ B and B ⊆ C .starting point We must show that A ⊆ C .conclusion to be shown Let x be any element in A. start of an element proof Then x is in B. (a) . It follows that x is in C. (b) . Thus every element in A is in C. since x could be any element of A Therefore, A ⊆ C [as was to be shown].(c) .

To determine

Proof. Suppose A,B and C are sets and AB and BC. To show that AC, we must show that every element in (a) is in (b). But given any elemetn in A, that elemetn is in (c) (because AB ), and so that element is also in (d) (because (e) )Hence, AC.

### Explanation of Solution

Given information:

For all sets A,B and C, if AB and BC, then AC.

Concept used:

: Subset

Calculation:

Let A,B and C be are any three sets.

Prove that, if AB and BC, then AC.

According to the definition of subset, the statement AB means that for every element in A is in the set B.

Also, the statement BC means that for every element in B is in the set C.

Similarly, the statement A AC means that for every element in A is in the set C.

Proof.

Since AB, every element in A, that element is in the set B and so that element is also in C because BC.

Therefore AC.

The appropriate word to fill in the blank (a) is A.

The appropriate word to fill in the blank (b) is C.

The appropriate word to fill in the blank (c) is B.

The appropriate word to fill in the blank (d) is C.

The appropriate word to fill in the blank (e) is BC.

Conclusion:

The appropriate word to fill in the blank (a) is A

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