Question

Use the construction given in the proof of Theorem 1.25 to give the state diagrams of DFAs accepting the union of the languages described in a. Exercises 1.6a and 1.6b. b. Exercises 1.6c and 1.6f. 1.6 Give state diagrams of DFAs recognizing the following languages. In all parts, the alphabet is {0,1}.

a. {w w begins with a 1 and ends with a 0}
b. {wcontains at least three 1s}
c. {ww contains the substring 0101 (i.e., w = x0101y for some x and y)}
d. {w/w has length at least 3 and its third symbol is a 0}
e. {w/ w starts with O and has odd length, or starts with 1 and has even length}
f. {w/ w doesn't contain the substring 110}

Answer 1

The question deals with properties of sets in set theory whether certain operations involving sets A, B, and F imply that A equals B. Equality in products or combinations does not necessarily imply that A equals B, as factors like an empty set F or additional elements combined with F could affect the outcome.

Step-by-step explanation:

The question asks whether the equality of certain products or combinations of sets A, B, and F implies that A equals B. This is a question related to set theory and the algebraic properties of sets. Set theory is a fundamental part of modern mathematics, and understanding the properties and operations involving sets is crucial for exploring more complex mathematical concepts.

(a) If A × F = B × F, we cannot necessarily conclude A = B. The reason is that F could be an empty set, making A × F and B × F both empty regardless of whether A and B are equal or not.

(b) If A F∪ F, which seems to mean A combined with F equals B combined with F, then it also does not imply A = B. One set could contain elements not present in the other, but when each is combined with F, the resulting sets are equal.

(c) If FÅ = BF, where I presume Å implies some operation involving F, without further context it's difficult to say definitively whether A equals B. Different operations could lead to different conclusions; we need more information about the operation and the sets in question.