pleaz solve this urgently

1. [8 marks] (a) Write down the 2 x 2 matrices that represent the following linear transformations of the plane. Specify for each transformation which points the points (1, 0} and {0, 1) map to. Also make reasonably accurate drawn-[gs (by hand or using MATLAB) of the image of the (rst quadrant) unit square [:11 under each transformation. You may use the MATLAB program given in Example 6.3 of the Unit Notes to create or check your drawings, which you can copy from the tile lintran .111 (available on the LMS). Also answer the questions. 1. A vertical shear T1 with shear factor 1. 2. The transformation T2 that maps point (0, 1) to {1/ (5., 1/~./) and (1, 0} to (1/\\/5_, I/). (see footnotel). Is this transformation a simple reection or a simple rotation? 3. The transformation T3 given by T3(x, y) = (ix + 393;, %x 393;). Is this transforma- tion a mirror reflection? If so, what is the reection line? (b} Create four diagrams that show the geometric transformation of the unit square D1 when applying the following transformations to it: T1, T2, T1 followed by T2, and T2 followed by T1. (c} Compute A121: and AgAl. Do the matrices A1 and A2 commute and is this in line with the result of part b? 2. [7 marks] Calculate by hand the appropriate truth table to prove the following logical equiva- lence (rVP)+q%v(PAra)A(r->q) Explain why your truth table shows that this proposition is valid. (As a voluntary additional exercise, you may also attempt to prove the logical equivalence by using the relations given in Tables 7.6-7.7 in the Unit Notes; however you do not need to submit this) 3. [5 marks] Decide whether or not the following propositions are true or false, where in each case the universal set is the set of positive integers NV\". The number x > 0 E R is an unspecied constant. Justify each of your conclusions with a proof or a counterexample. 1My apologies, an earlier versioni incorrectly said (1,0) to ( 1 / J5. 14/5) omitting a '/ ' sign in the y-coordinate. (a) as" (411 = TE) (b) an (HI/3 = 32) (C) W: (2\" 2 Hz) (a) 3n ((x+3) =2; 0. ,'._.(::'),x\""'2") 4. [5 marks] Does the following equality hold in every Boolean algebra? Show all working [2 (x+y'})= ()z+x(z+z}(z+y)