a) Make a table showing the truth value of the predicate p(x, y) = xy ≥ 0 for all possible values of x, y ∈ {−1, 0, 1}. (1 mark) b) Is ∀x∃y p(x, y) true?
c) Is ∃x∀y p(x, y) true?
a) Truth table for all values of x & y.........
Sr.No. | x | y | xy |
1 | -1 | -1 | 1 |
2 | -1 | 0 | 0 |
3 | -1 | 1 | -1 |
4 | 0 | -1 | 0 |
5 | 0 | 0 | 0 |
6 | 0 | 1 | 0 |
7 | 1 | -1 | -1 |
8 | 1 | 0 | 0 |
9 | 1 | 1 | 1 |
extracting values for p(x, y) = xy ≥ 0
Sr.No. | x | y | xy |
1 | -1 | -1 | 1 |
2 | -1 | 0 | 0 |
4 | 0 | -1 | 0 |
5 | 0 | 0 | 0 |
6 | 0 | 1 | 0 |
8 | 1 | 0 | 0 |
9 | 1 | 1 | 1 |
b) Is ∀x∃y p(x, y) true?
False
for x=0 there is no 'y' which yields xy=1.
c) Is ∃x∀y p(x, y) true?
False
for y=0 there is no 'x' which yields xy=1.
a) Make a table showing the truth value of the predicate p(x, y) = xy ≥...
d) Let F(x, y)-xy'+x'y. a) 2. Construct a truth table for F. ND, OR, and NOT gates. b) Design a circuit with inputs x and y to implement F(x, y) using only AND, OR c) Use DeMorgan's law to find the complement of F, ie, find F'(x, y). d) Show that F'(x,x)-1.
Given the function F(x,y) = y'+(x+y) : a) Make a truth table for F. [4 marks] b) Express F as a sum of products. [3 marks] c) Simplify F, either algebraically or by an explanation based upon the truth table. [3 marks]
question 5.20. Let P(x, y) be the predicate x + y = 10 where x and y are any real numbers. Which of the following statements are true? (a) (∀x)(∃y), P(x, y). (b) (∃y)(∀x), P(x, y)
6. Consider the predicates M(x), F(x), and P(x, y) in a domain of people. The predicate M(x) states of a person that he is male, the predicate F(x) states of a person that she is female, the predicate P(z, y) states that x is the parent of y. Write the following queries in the Predicate Logic. (a) Find the people who are mothers (b) Find the people who do not have an uncle.
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