1. Predicates and Quantifiers (7 points) (1) (3 points) Let L( denote the statem
ID: 3879192 • Letter: 1
Question
1. Predicates and Quantifiers (7 points) (1) (3 points) Let L( denote the statement "x has used LaTeX." and D denote the domain of a students in our class. Express each of these quantified propositions in English: (a) VED : L(x) (2) (2 points) Let Q(x) denote the statement "2 > 3r" and Z denote all integers (i.e., -3,-2,-1,0,1,2,3,..). Determine the truth value of each of these statements (a) Q(2) (b) Q(4) (3) (2 points) Translate the following statements to English where B(r) is "r understands boolean algebra" and M(x) is "r has taken discrete math" and the domain D is all students at UTSA (a) VIED : (M(z) B(z))Explanation / Answer
In Discrete Math:
The upside down inverted A is called the universal quatinfier it means "For all".
The sideways inverted E is called the existential quantifier. It means "There exists".
1) a. All students in our class have used LaTeX.
b. There does not exist any student in our class who (No student in our class) has used LaTeX.
c. There are some students in our class who have NOT used LaTeX.
2) a. Q(2) equates to 2^2 > 3*2 which is FALSE as 4 < 6.
b. Q(4) equates to 2^4 > 3*4 which is TRUE as 16 > 12
c. This says: For all integers x belongs to Z such that "2^x > 3*x OR x<4" is true. This is TRUE as Q(x) is true for for all x >= 4 and it is false when x<4 but then the second statement "x < 4"stands true. And sincce, in OR either of the statements needs to be true for the whole proposition to be true.
d. This translates to : There exists an integer x belongs to Z for which "2^x > 3*x AND x>4" is true. This is FALSE as it is the exact compliment of the above proposition in (c). In an AND operation both statements needs to be true for whole proposition to be true. There is no value for x for which this can happen.
3) a. All students at UTSA have taken discrete math which implies they understand boolean algebra.
b. There are some students at UTSA who understand booleran algebra AND have NOT taken discrete math.
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