Search & Filtering
What will be the inverse of the following functions from R to R?
(a)
f: R—>R defined by f(x) = x
(b)
f: R—>R defined by f(x) = x + 1
(c)
f: R—>R defined by f(x) = – 3x+4
(d)
f: R—>R defined by f(x) = x^3
(e)
f: R—>R defined by f(x) = sin (x)
Draw graphs of the following functions.
(a)
f: R—>R defined by f(x) = x
(b)
f: R—>R defined by f(x) = |x|
(c)
f: R—>R defined by f(x) = x + 1
(d)
f: R—>R defined by f(x) = – 3x+4
(e)
f: R—>R defined by f(x) = Floor(x)
(f)
f: R—>R defined by f(x) = Ceiling(x)
(g)
f: R—>R defined by f(x) = x2
(h)
f: R—>R defined by f(x) = x3
Find out which of the following functions from R to R are (i) One-to-one, (ii) Onto, (iii) One-to-one correspondence.
(a)
f: R—>R defined by f(x) = x
(b)
f: R—>R defined by f(x) = |x|
(c)
f: R—>R defined by f(x) = x + 1
(d)
f: R—>R defined by f(x) = x2
(e)
f: R—>R defined by f(x) = x3
(f)
f: R—>R defined by f(x) = x – x2
(g)
f: R—>R defined by f(x) = Floor(x)
(h)
f: R—>R defined by f(x) = Ceiling(x)
(i)
f: R—>R defined by f(x) = – 3x+4
(j)
f: R—>R defined by f(x)= – 3x2 +7
State which of the following are not a function from R to R and why.
(a)
f(x) = 1/x
(b)
f(x) = 1/(1+x)
(c)
f(x) = (x)½
(d)
f(x) = ±(x2+1)½
(e)
f(x) = sin(x)
(f)
f(x) = ex
What are the truth sets of the predicates P(x), Q(x), and R(x), where the domain is the set of integers and 1. P(x) is “|x| = 4,” 2. Q(x) is “x2 = 16,” 3. R(x) is “|x| = x”
£(x - 50) =60,£(y-20) = -20,£(x -50)^2 = 450,£(y - 20)^2 = 200,£(x - 50)(y - 20) = -100
Find the simplest form for the following boolean expressions:
1) (A.B'.C')+(A'.B'.C')+(A'.B.C')+(A'.B'.C)
2) (A'.B.C)+(A'.B.C)+(A.B.C')+(A.B'.C')+(A'.B.C')+(A'.B'.C')
3) (A+B+C)(A+B'+C')(A+B+C')(A+B+C')
( ' = Not )
(Direct proof) A claim is given as a quantified statement: “The product of two odd numbers is an odd number” a) (1 point) Write the domain of the variables: b) (4 points) Write the statement using quantifiers and an implication of propositional functions: c) (15 points) Prove the statement by direct proof (Assume the hypothesis and derive the conclusion)
1. What are the converse, inverse and contrapositive of the statement
“If you try hard, then you will win”
Converse
Inverse
Contrapositive
b. Determine whether this proposition is a tautology, contradiction or contingency using the truth table: (p q) (q → rp) (p r)
Question 4
a. Write the set S= {1, 3, 5, 7} in set builder notation.
b. Write the set T={ x | x ∈ R, x2 = 9 or x2 = 64} in a list form.
c. Let A = {x, y, z }and B = {2, 5, 6}
List the elements of the following sets.
i. Power(B ) =
ii. B A =
d. What are the truth sets of the predicates P(x), Q(x), and R(x), where the domain is the set of integers and
1. P(x) is “|x| = 4,”
2. Q(x) is “x2 = 16,”
3. R(x) is “|x| = x”
Question 5:
Given the below premises Prove: B.
1. (A v B) → C 2. A
3. C
Question 1: (2 marks) (C1)
Let’s consider a propositional language where
A =“I study data structure”,
B =“I study Bioinformatics”,
C =“I study programming”,
D =“I study discrete math”
a. “I study Bioinformatics if I study programming and discrete math”
b. “I cannot study data structure when I do not study programming or discrete math ”
c. “I study data structure if and only if I do not study Bioinformatics”
Question 2: (2 marks)
Let’s consider a propositional language where
p means “Paola is happy”,
q means “Paola paints a picture”,
r means “Renzo is happy”.
Write English statement that corresponds to the following compound propositions:
1. p ∧ q → ¬ r
2. ¬ (p ∧ ¬q)
