Suppose A,B,C are sets. Show that if ∣A∣≤∣B∣ and ∣B∣≤∣C∣, then ∣A∣≤∣C∣.
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Answer from Sia
Posted 26 days ago
Solution by Steps
step 1
To show that if ∣A∣≤∣B∣ and ∣B∣≤∣C∣, then ∣A∣≤∣C∣, we can use the transitive property of inequalities
step 2
Since ∣A∣≤∣B∣, there exists an injection from set A to set B. Similarly, since ∣B∣≤∣C∣, there exists an injection from set B to set C
step 3
By composing these injections, we can construct an injection from set A to set C, which implies that ∣A∣≤∣C∣
Answer
∣A∣≤∣C∣
Key Concept
Transitive property of set sizes (cardinalities)
Explanation
The transitive property allows us to conclude that if one set can be injected into another, and that set can be injected into a third, then the first set can also be injected into the third, establishing the inequality of their sizes.
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