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== Proving one set is a subset of another set == | == Proving one set is a subset of another set == | ||
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Given sets A and B we say that is is a subset of B if every element of A is also an element of B, that is, | Given sets A and B we say that is is a subset of B if every element of A is also an element of B, that is, | ||
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'''·''' Suppose x ∈ A | '''·''' Suppose x ∈ A | ||
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1. Say what it means for x to be in A | 1. Say what it means for x to be in A | ||
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3. Conclude that x satisfies what it means to be in B | 3. Conclude that x satisfies what it means to be in B | ||
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'''·''' Conclude x∈B | '''·''' Conclude x∈B | ||
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'''·''' Suppose x ∈ A: | '''·''' Suppose x ∈ A: | ||
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1. What it means for x to be in A: x = 6k for any scalar k | 1. What it means for x to be in A: x = 6k for any scalar k | ||
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3. What it means for x to be in B: x = 2C | 3. What it means for x to be in B: x = 2C | ||
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'''·''' Conclude x∈B | '''·''' Conclude x∈B | ||
Revision as of 08:00, 25 November 2012
Proving one set is a subset of another set
Given sets A and B we say that is is a subset of B if every element of A is also an element of B, that is,
x∈A implies x∈B
Basic Outline of the Proof that A is a subset of B:
· Suppose x ∈ A
1. Say what it means for x to be in A
2. Mathematical details
3. Conclude that x satisfies what it means to be in B
· Conclude x∈B
Example
Let A be the set of scalars divisible by 6 and let B be the even numbers. Prove that A is a subset of B.
· Suppose x ∈ A:
1. What it means for x to be in A: x = 6k for any scalar k
2. x = 2 × (3k)
3k = C
3. What it means for x to be in B: x = 2C
· Conclude x∈B
