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 ) .

Q: Expert Answer

To determine Proof. Suppose A , B and C are sets and A ⊆ B and B ⊆ C . To show that A ⊆ C , we must show that every element in ( a ) is in ( b ) . But given any elemetn in A , that elemetn is in ( c ) (because A ⊆ B ), and so that element is also in ( d ) (because ( e ) )Hence, A ⊆ C . Explanation Given information: For all sets A , B and C , if A ⊆ B and B ⊆ C , then A ⊆ C . Concept used: ⊆ : Subset Calculation: Let A , B and C be are any three sets. Prove that, if A ⊆ B and B ⊆ C , then A ⊆ C . According to the definition of subset, the statement A ⊆ B means that for every element in A is in the set B . Also, the statement B ⊆ C means that for every element in B is in the set C . Similarly, the statement A A ⊆ C means that for every element in A is in the set C . Proof. Since A ⊆ B , every element in A , that element is in the set B and so that element is also in C because B ⊆ C . Therefore A ⊆ C . 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 B ⊆ C . Conclusion: The appropriate word to fill in the blank ( a ) is A

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Discrete Mathematics With Applicat...

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Publisher: Cengage Learning,

ISBN: 9781337694193

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Answer 3

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Answer 4

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Answer 5

To determine

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

Explanation of Solution

Given information:

For all sets A,B and C, if A⊆B and B⊆C, then A⊆C.

Concept used:

⊆: Subset

Calculation:

Let A,B and C be are any three sets.

Prove that, if A⊆B and B⊆C, then A⊆C.

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

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

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

Proof.

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

Therefore A⊆C.

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 B⊆C.

Conclusion:

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

Discrete Mathematics With Applicat...

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Discrete Mathematics With Applicat...

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Statement	Explanation
Suppose A, B, and C are any sets such that