I've got this problem about sets, and cardinality. I don't really understand it other than cardinality is the number of elements within each set, I don't understand a lot of the signs used within the sets, if anyone can answer this question, explaining the reasoning behind their answer I would be very greatful.
I understand this is strictly a homework question, but I figured I'd ask here before going into my professors office hours. Thanks in advance.
• S = {s ∈ Z+∣∃y ∈ Z+, s + y = 25}
• T = {t ≤ 36∣∃x ∈ Z+, t = x 2}
• M = {m ∈ Z+∣m ≤ 8}
• W = P({a, b, c}), namely the powerset of the set {a, b, c}
• X = {(a1, a2, a3, a4) ∈ Z+ × Z+ × Z+ × Z+∣(∀i, ai < 5) ∧ (∀i, j, i ≠ j → ai ≠ aj)}
• Y = {(a1, a2, a3) ∈ Z+ × Z+ × Z+∣(∀i, ai < 4) ∧ (∀i, j, i ≠ j → ai ≠ aj)}
Moreover, denote the set containing these sets as E = {S, T,M,W, X, Y } (1) Note that E is not a union of the sets it contains.
In each of the following questions, you must justify your answer. This justification may consist of either listing all out all of the elements of the set concerned, or a proof by another means.
a. What is the cardinality of S? Provide justification.
b. What is the cardinality of T? Provide justification.
c. What is the cardinality of W? Provide justification.
d. What is the cardinality of X? Provide justification.
e. What is the cardinality of Y ? Provide justification
cardinality of S is 24, since s+y =25 , where s and y are positive integers then the least possible value of y is 1 and the maximum value is 24. the same goes with x. thus the cardinality of S 1s 24 [1,24]
Cardinality of T is 18, since t=2x and x is positive then t is the set positive even numbers from 2 to 36. and there are 18 positive even integers in [2,36]
the cardinality of M is obviously 8, since m are positive integers less than or equal to 8.
cardinality of W is 8, since there are 3 elements (namely a,b,c) then it's power set is 2*3=8 if a set has n elements then the cardinality of it's power set is 2*n
i just dont unerstand some notations in X and Y sorry.