Search & Filtering
Suppose U = {1, 2, 3, 4, 5, a, b, c} is a universal set with the subset A = {a, b, c, 1, 2, 3, 4}. Answer questions 1 and 2 by using the given sets U and A.
Question 1 Which one of the following relations on A is NOT functional?
1. {(1, 3), (b, 3), (1, 4), (b, 2), (c, 2)}
2. {(a, c), (b, c), (c, b), (1, 3), (2, 3), (3, a)}
3. {(a, a), (c, c), (2, 2), (3, 3), (4, 4)}
4. {(a, c), (b, c), (1, 3), (3, 3)}
Question 2
Which one of the following alternatives represents a surjective function from U to A?
1. {(1, 4), (2, b), (3, 3), (4, 3), (5, a), (a, c), (b, 1), (c, b)}
2. {(a, 1), (b, 2), (c, a), (1, 4), (2, b), (3, 3), (4, c)}
3. {(1, a), (2, c), (3, b), (4, 1), (a, c), (b, 2), (c, 3)}
4. {(1, a), (2, b), (3, 4), (4, 3), (5, c), (a, a), (b, 1), (c, 2)}
How many elements does A=0 have ?
Question 5. A graph has 24 edges, 4 vertices of degree 5, and all other vertices of degree 2.
How many nodes does it have in total?
A committee of 3 individuals decides on issues for an organization. Each individual votes either a YES or a NO for each proposal to pass. A proposal is passed if it receives at least two YES votes . Design a circuit that determines if the proposal will pass
Let R be the relation on Z
(the set of integers) defined by
(x, y) R iff x
2 + y2 = 2k for some integers k 0.
Answer questions 13 to 15 by using the given relation R.
Question 13
Which one of the following is an ordered pair in R?
1. (1, 0)
2. (2, 9)
3. (3, 8)
4. (5, 7)
Question 14
R is symmetric. Which one of the following is a valid proof showing that R is symmetric?
1. Let x, y Z be given.
Suppose (x, y) R
then x
2 + y2 = 2k for some k 0.
ie y
2 + x2 = 2k for some k 0.
thus (x, y) R.
2. Let x, y Z be given.
Suppose (x, y) R
then x
2 + y2 = 2k for some k 0.
ie y
2 + x2 = 2k for some k 0.
thus (y, x) R.
3. Let x, y Z
be given.
Suppose (x, y) R
then x
2 + y2 = 2k for some k 0.
thus (y, x) R.
4. Let x, y Z be given.
Suppose (x, x) R
then x
2 + y2 = 2k for some k 0.
ie y
2 + x2 = 2k for some k 0.
thus (y, y) R.
Prove that if n is an integer and 3n + 2 is even, then n is even using
a) a proof by contraposition.
b) a proof by contradiction
only if you work attentively in the kitchen. Having no Sui gas is a sufficient condition for you to
not get good food. If the weather is too hot, then you will not work attentively in the kitchen. The
weather is not too hot and there is no Sui gas.” Show complete working of your solution.
1. ∼ (p ∧ q) and ∼ p∨ ∼ q (de Morgan’s Laws)
2. ∼ (p ∧ q) and ∼ p∨ ∼ q
3. (∼ p ∨ q) ∧ (∼ q) ≡∼ (p ∨ q).
4. (p∧ ∼ q) ∨ (∼ p ∨ q) ≡ t.
1. p∨ ∼ q
2. p ∨ (q ∧ r )
3. (p ∨ q) ∧ (p ∨ r )
You visit an island where three triplet brothers named Lanister, Lewis and Tom, live. They are indistinguishable in appearance, but Lanister and Lewis, are knaves whereas Tom is a knight. One day you meet one of the three on the street and wish to know whether or not he is Lewis, because Lewis owes you money. You are allowed to ask him only one yes/no question (your question may not have more than three words!) What question would work? Provide logical reasoning for your answer.
