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Find a counterexample, if possible, to these universally quantified statements, where the domain for all variables consists of all real numbers.
a) ∀x(x2 ( x)
b) ∀x(x2 ( 2)
c) ∀x(|x| > 0
Determine the validity of the following argument.
“To pass the Discrete Mathematics, it is necessary to pass both the course work and
the final examination. Either John will have to work hard in Discrete Mathematics or he will fail his final examination. However, John will not both work hard in Discrete Mathematics and pay attention in lectures. Therefore, if John pays attention in lectures, then he will pass the Discrete Mathematics.”
In the given picture are three men: Neil Armstrong, Michael Collins and Buzz
Aldrin. They were on the Apollo 11 that set the first man on the moon in 1969.
Neil Armstrong was the first man walking on the moon. Which is an example of
an ordinal number?
(2)
A. Three
B. 11 (as in Apollo 11)
C. First
D. 1969
E. None of the above
2. Use set builder notation to give a description of each of these sets. a) {0, 3, 6, 9, 12} b) {−3, −2, −1, 0, 1, 2, 3} c) {m, n, o, p}
Let S = {Barnsley, Manchester United, Southend, Sheffield United, Liverpool, Maroka Swallows, Witbank Aces, Royal Tigers, Dundee United, Lyon} be a universal set, A = {Southend, Liverpool, Maroka Swallows, Royal Tigers}, and B = {Barnsley, Manchester United, Southend}. Find n((A ∩ B)').
Thanks..
g:X-Y be defined by g(x) = (5x-3)/2
Find:
1. f(-5/2)
2. f(n)
3. g(1)
4. fo g (x)
Consider the following functions and determine if they are bijective. [A function is said to be bijective or bijection, if a function f: A→B is both one-to-one and onto.]
(a) f: Z × Z→Z, f(n, m) = n2 + m2
(b) f: R→R, f(x) = x3 − 3
(c) f: R × R→R, f(n, m) = 2m − n
Give an example of two uncountable sets A and B with a nonempty intersection, such that A−B is
(a) Finite
(b) Countably infinite
(c) Uncountably infinite
Given two sets A and B, for each of the following statements, what can you conclude about the sets?
For example: consider the statement A−B=∅, this could be possible if -
Scenario - 1: if A=B then A−B=∅
Scenario - 2: Since A−B=A−(A∩B), if (A∩B) = A then A−B=∅
Scenario - 3: A=∅ in which case, no matter what B is A−B=∅.
Therefore we can conclude that if A−B=∅, then one of the above scenarios must be true. You do not need to draw exactly 3 conclusions. Try to answer with an exhaustive list of conclusions you can draw from each of the following statements:
(a) A∪B=A
(b) A∩B=A
(c) A−B=A
(d) A∩B=B∩A
(e)A−B=B−A
#339914 on Dec 2023