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. 设集合

A=\left\{a_{1}, a_{2}, a_{3}, a_{4}\right\}

，若集合

A

中所有三个元素的子集中的三个元素之和组...

Mar 8, 2024

. 设集合

A=\left\{a_{1}, a_{2}, a_{3}, a_{4}\right\}

，若集合

A

中所有三个元素的子集中的三个元素之和组成的集合为

B=\{2,3,4,6\}

，则集合

A=

Solution by Steps

step 1

To find the set

A

, we need to solve the system of equations given by the sums of the subsets of three elements each

step 2

The system of equations is:

a1 + a2 + a3 = 2

a1 + a2 + a4 = 3

a1 + a3 + a4 = 4

a2 + a3 + a4 = 6

step 3

Solving this system of equations, we find that

a1 = -1

a2 = 1

a3 = 2

, and

a4 = 3

step 4

Therefore, the set

A

\{-1, 1, 2, 3\}

Answer

A = \{-1, 1, 2, 3\}

Key Concept

Solving a system of linear equations

Explanation

The solution to the system of equations represents the elements of set

A

such that the sums of all possible three-element subsets match the elements of set

B