Let U be the universe and let A and B be subsets of U. Prove that A(union) B= {e
ID: 3081653 • Letter: L
Question
Let U be the universe and let A and B be subsets of U. Prove that A(union) B= {empty set} if and only if
A (subset) B(complement).
Explanation / Answer
Wemustprove: (?y ? R) (y ? A ? y ? B) . Form of Proof: • Let y = b where b is a specific real number we have derived. • Verify that b ? A. • Verify that b ? B. As is often the case, the second and third steps in the form above are proofs within a proof, so we will expand our form to include each of these. To show b ? A we must prove (?x ? R) f(x) = b. To show that b ? B we must show (?x ? R) g(x) = b. Proof by contradiction often works well for statements of this sort; that is, statements that something doesn’t happen. We now repeat the form of the proof, with more detail: 7 Form of Proof: • Let y = b where b is a specific real number we have derived. • Verify that b ? A. • Set x = a where a is a specific real number we have derived. • Verify that f(a) = b. • Verify, by contradiction, that b ? B. • Assume b ? B. • Use the definition of B to expand on the previous assumption. • Give a logical argument that leads to a contradiction. Exercise: Follow the form above to write a proof that A ? B. Proper Subsets: Definition 3: ForsetsA and B in the universal set U we say that A is a proper subset of B, written A ? B, provided A ? B and A = B. Equivalently, A ? B, provided A ? B and B ? A. Thus, in symbols, A ? B means: (?x ? U) (x ? A ? x ? B) ? (?y ? U) (y ? B ? y ? A) . To Prove: A ? B Form of Proof: First, prove that A ? B. • Let x be arbitrary (variable) in U. (If it contributes, expand on what it means to be in U.) • Suppose x ? A. (If it helps, expand on what x ? A means.) • Now give a logical argument that concludes with x ? B. Now show that B ? A. • Let y = b, where b is a specific element in U. • Verify that b ? B; and • Verify that b ? A. Example 3 (cf. Example 2): Let A = { x ? R| x2 - 4 > 0 } and let B = { x ? R| x2 - 9 > 0 }. Prove that B ? A. Proof: ToseethatB ? A, let x ? R and suppose x ? B. Then x2 - 9 > 0. Now x2 - 4 = (x2 - 9) + 5. Since both x2 - 9 and 5 are positive, it follows that their sum is positive; that is, x2 - 4 > 0. Therefore, x ? A. To see that A ? B (as in Example 2) let x = 3. Then x2 - 4 = 32 - 4 = 9 - 4 = 5 and 5 > 0, so 3 ? A. But x2 - 9 = 32 - 9 = 9 - 9 = 0, so 3 ? B. 8 This proves that B ? A. Exercise 5: LetM2(R) denote the set of all 2 × 2 matrices with real number entries, let O2 = 0 0 0 0 , and let C = 1 2 3 6 . Now set S = {A ? M2(R) |A = -3a a 0 0 where a ? R} and let T = {B ? M2(R) |BC = O2 }. Prove that S ? T
Related Questions
Navigate
Integrity-first tutoring: explanations and feedback only — we do not complete graded work. Learn more.