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Idea the definition this expresses cisely Definition is Similarly, {x,y} Definition 15. , y}. For example, Snork, Mickey Mouse} the set consisting exactly of the three individuals, Alice Zilch, Sam Snork and Mickey Mouse. Now it may very well happen that a given object or individual has different names. " does the symbol {Pierpont, Stinky} denote? " But "Pierpont" and "Stinky" are different names for the same individual; containing the individuals consequently = {Pierpont, Stinky} More formally, if x = y = {x} } Similarly, if x = z and v {x,y,z,v,w} = w = = {x,v} = Verify the foregoing equations.

Prove: If A C 5 and 4. Prove: If A C B and A 5. Is it true that ment but 6. is false, A (2 C then if A C -B J5 C ACB ( CA and - C then A C C. C then B(C. and B give examples in C < which A

Thus, by Rule I, p or q and q or p are equivalent, and so are the sentences p and q and q and p. are equivalent. by Rule Rule III: II, If a sentence is built and, and not, and then the result is if from parts any part is p, q, r, . . by means of or, replaced by an equivalent sentence, a sentence equivalent with the original one. THE ELEMENTS OF THE THEORY OF SETS EXAMPLES 1. The sentences (p and q) or r and (q and p) or r are equivalent, since the second is obtained from the first by replacing the part p and 2.

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3-D Designs

by David

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