Search & Filtering
Let A={a, b, c}, B={d, e}, C={a, d}.
Find (i) A × B (ii) B × A (iii) A × (B C)
(iv) (A C) x B (v) A ∆ (B – C) (vi) Complement of (A ∆ B)
(vii) P(B C)
Let R ⊆ S × S be an equivalence relation on a set S. For an element x ∈ S, let
S(x) = {y ∈ S : (x, y) ∈ R}. Show that for every pair of elements x, y ∈ S, either
S(x) = S(y) or S(x) ∩ S(y) = ∅.
In the game of poker, five cards are drawn from a deck of 52 cards. A set of 5 cards is
said to be a hand. [A standard 52-card deck contains four suits of 13 cards each. The
suits are spades (♠), clubs (♣), hearts (♥) and diamonds(♦). The 13 cards in each
suit are labeled A,2,3,. . . ,10,J,Q,K.]
(i) How many hands contain Four Of a Kind, i.e. four cards with the same value, four
eg: four As, four 7s etc.
(ii) How many hands form a Straight, i.e. 5 cards in increasing order such that not all
are from the same suit? The lowest card is A although a Straight can also end in an
A.
[Eg: A♣, 2♠,3♠,4♥,5♦ and 10♦, J♠,Q♥,K♥,A♠ ]
(iii) How many hands have Two Pairs, i.e. two cards with one value, two cards with
another value and a fifth with a different value?
[Eg: 2♠, 2♦, J♣, J♠, 5♥]
Rahul left his cycle in a parking lot that is filled with bikes and cycles, parked neatly
in several rows. When he returns, he notices someone push all the vehicles on the
left end of each row, causing them to fall. Rahul knows that if a bike falls, then the
vehicle to its immediate right will also fall, but if a cycle falls, then the next vehicle
will fall only if that is also a cycle. Rahul hasn’t spotted his cycle yet, but he deduces
by mathematical induction that it will fall.
“Let P(n) statement: ‘In every row, every cycle in nth position will fall.’ I
know that P(1) is true - all cycles in the first position were pushed. For the inductive
step, consider any row with a cycle in the nth position. By the inductive assumption,
this cycle will fall. Now consider the next vehicle, which is in the (n + 1) th position.
If it is a bike, I don’t have to prove anything; if it is a cycle, it will fall because it was
pushed by the cycle on its left. ” What is the
flaw in his argument?
For each proposition below, decide whether it is true or false and give a brief explanation. Assume the universe (domain of variables) to be Z, the set of integers.
(1) ((x = 5) ∧ (y = 1)) → ((x > 10) ∨ (y > 0))
(2) ∀x((x < 0) ∨ (x^2 ≥ x))
(3) ¬(∃xP(x)) ↔ (∀x¬P(x)) for all predicates P(x)
Using a truth table, prove or disprove the following:
~[(~p∧q)↔ r]≡ ~(p∨~q)↔ ~r
Let A = { 0,2,4,6,8,10 }, B = { 0,1,2,3,4,5,6 }, and C = { 4,5,6,7,8,9,10 }. (2 pts each)
Find
a) A ∩ B ∩ C
b) A ∪ B ∪ C
c) ( A ∪ B) ∩ C
d) ( A ∩ B) ∪ C
Determine whether each of these pairs of sets are equal. (1 pt each)
a) {1,3,3,3,5,5,5,5,5}, {5,3,1}
b) {{1}}, {1,{1}}
c) ∅, {∅}
Define a relation T from R to R as follows: For all real numbers x and y
(X,y) E T means that y^2- x^2= 1.
Is T a function? Explain
Give the truth value.
P-~q,if q-~p is false
