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22 Cards in this Set
- Front
- Back
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Subset
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A is a _______ of B only if every element of A is also an element of B.
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proper subset
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A is a ______ ______ of B only if every element of A is in B but there is at least one element of B that is not in A.
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A = B if...
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A _____ B only if every element of A is in B and every element of B is in A.
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The union of A and B
Let A and B be subsets of a universal set U. |
The set of all elements x in U such that x is in A or x is in B.
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The intersection of A and B
Let A and B be subsets of a universal set U. |
The set of all elements x in U such that x is in A and x is in B.
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The difference of A-B
(Or relative complement of B in A) Let A and B be subsets of a universal set U. |
The set of all elements x in U such that x is in A and x is not in B.
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The complement of A
Let A and B be subsets of a universal set U. |
The set of all elements x in U such that x is not in A.
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∼(p ∧ q) ≡ ....
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.... ≡ ∼p ∨ ∼q
The negation of an and statement is logically equivalent to the or statement in which each component is negated. |
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∼(p ∨ q) ≡ ....
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.... ≡ ∼p ∧ ∼q
The negation of an or statement is logically equivalent to the and statement in which each component is negated. |
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p →q
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It is false when p is true and q is false;
*otherwise it is true* |
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p →q ≡ ....
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.... ≡ ∼p ∨ q
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∼(p →q) ≡ ....
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.... ≡ p ∧ ∼q
The negation of “if p then q” is logically equivalent to “p and not q. |
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The contrapositive of p →q is ....
∀x ∈ D, if P(x) then Q(x). A conditional statement is logically equivalent to its contrapositive. |
.... of p →q is ∼q →∼p.
.... ∀x ∈ D, if ∼Q(x) then ∼P(x). |
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The converse of p →q is ....
∀x ∈ D, if P(x) then Q(x). 1. A conditional statement and its converse are not logically equivalent. 3. The converse and the inverse of a conditional statement are logically equivalent to each other. |
The.... of p →q is q → p,
∀x ∈ D, if Q(x) then P(x) |
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The inverse of p →q is ....
∀x ∈ D, if P(x) then Q(x). 2. A conditional statement and its inverse are not logically equivalent. 3. The converse and the inverse of a conditional statement are logically equivalent to each other. |
The .... of p →q is ∼p →∼q
∀x ∈ D, if ∼P(x) then ∼Q(x) |
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biconditional of p and q is ...
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“p if, and only if, q” "p ↔ q"
It is true if both p and q have the same truth values and is false if p and q have opposite truth values. |
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∼(∀x ∈ D, Q(x)) ≡ ....
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.... ≡ ∃x ∈ D such that ∼Q(x).
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∼(∃x ∈ D such that Q(x)) ≡ ....
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.... ≡ ∀x ∈ D,∼Q(x).
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∼(∀x, if P(x) then Q(x)) ≡ ....
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.... ≡ ∃x such that P(x) and ∼Q(x).
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Even and Odd
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Symbolically, if n is an integer, then
n is even ⇔ ∃an integer k such that n = 2k. n is odd ⇔ ∃an integer k such that n = 2k + 1. |
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Prime/Composite
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In symbols:
n is prime ⇔ ∀positive integers r and s, if n = rs then either r = 1 and s = n or r = n and s = 1. n is composite ⇔ ∃positive integers r and s such that n = rs and 1 < r < n and 1 < s < n. |
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Proofs
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1. Copy the statement of the theorem to be proved on your paper.
2. Clearly mark the beginning of your proof with the word Proof. 3. Make your proof self-contained. 4. Write your proof in complete, gramatically correct sentences. 5. Keep your reader informed about the status of each statement in your proof. 6. Give a reason for each assertion in your proof. 7. Include the “little words and phrases” that make the logic of your arguments clear 8. Display equations and inequalities. |
